Formula.doc

7/29/2019 Formula.doc

1/215

Wiki Loves Monuments: Photograph a monument, help

Wikipedia and win!

List of trigonometric identitiesFrom Wikipedia, the free encyclopedia

Cosines and sines around theunit circle

Trigonometry

History

Usage

Functions

Generalized

Inverse functions

Further reading
http://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/History_of_trigonometryhttp://en.wikipedia.org/wiki/History_of_trigonometryhttp://en.wikipedia.org/wiki/Uses_of_trigonometryhttp://en.wikipedia.org/wiki/Uses_of_trigonometryhttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Generalized_trigonometryhttp://en.wikipedia.org/wiki/Generalized_trigonometryhttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_basic_trigonometry_topicshttp://en.wikipedia.org/wiki/List_of_basic_trigonometry_topicshttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svghttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_basic_trigonometry_topicshttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Generalized_trigonometryhttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Uses_of_trigonometryhttp://en.wikipedia.org/wiki/History_of_trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Unit_circlehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnoticehttp://www.wikilovesmonuments.in/?pk_campaign=Centralnotice


2/215

Reference

Identities

Exact constants

Trigonometric tables

Laws and theorems

Law of sines

Law of cosines

Law of tangents

Law of cotangents

Pythagorean theorem

Calculus

Trigonometric substitution

Integrals of functions

Derivatives of functions

Integrals of inverse functions

V

T

E

Inmathematics,trigonometric identities are equalities that involvetrigonometric functionsand

are true for every single value of the occurringvariables. Geometrically, these

areidentitiesinvolving certain functions of one or moreangles. They are distinct fromtriangle

identities, which are identities involving both angles and side lengths of atriangle. Only the

former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be

simplified. An important application is theintegrationof non-trigonometric functions: a common
http://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Exact_trigonometric_constantshttp://en.wikipedia.org/wiki/Trigonometric_tableshttp://en.wikipedia.org/wiki/Trigonometric_tableshttp://en.wikipedia.org/wiki/Law_of_sineshttp://en.wikipedia.org/wiki/Law_of_sineshttp://en.wikipedia.org/wiki/Law_of_cosineshttp://en.wikipedia.org/wiki/Law_of_cosineshttp://en.wikipedia.org/wiki/Law_of_tangentshttp://en.wikipedia.org/wiki/Law_of_tangentshttp://en.wikipedia.org/wiki/Law_of_cotangentshttp://en.wikipedia.org/wiki/Law_of_cotangentshttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functionshttp://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functionshttp://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Template:Trigonometryhttp://en.wikipedia.org/wiki/Template:Trigonometryhttp://en.wikipedia.org/wiki/Template_talk:Trigonometryhttp://en.wikipedia.org/wiki/Template_talk:Trigonometryhttp://en.wikipedia.org/w/index.php?title=Template:Trigonometry&action=edithttp://en.wikipedia.org/w/index.php?title=Template:Trigonometry&action=edithttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Identity_(mathematics)http://en.wikipedia.org/wiki/Identity_(mathematics)http://en.wikipedia.org/wiki/Identity_(mathematics)http://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Integralhttp://en.wikipedia.org/wiki/Trianglehttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Trigonometry#Triangle_identitieshttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Identity_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/w/index.php?title=Template:Trigonometry&action=edithttp://en.wikipedia.org/wiki/Template_talk:Trigonometryhttp://en.wikipedia.org/wiki/Template:Trigonometryhttp://en.wikipedia.org/wiki/List_of_integrals_of_inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_integrals_of_trigonometric_functionshttp://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/Calculushttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Law_of_cotangentshttp://en.wikipedia.org/wiki/Law_of_tangentshttp://en.wikipedia.org/wiki/Law_of_cosineshttp://en.wikipedia.org/wiki/Law_of_sineshttp://en.wikipedia.org/wiki/Trigonometric_tableshttp://en.wikipedia.org/wiki/Exact_trigonometric_constants


3/215

technique involves first using thesubstitution rule with a trigonometric function, and then

simplifying the resulting integral with a trigonometric identity.

Contents

[hide]

1 Notation

o 1.1 Angles

o 1.2 Trigonometric functions

2 Inverse functions

3 Pythagorean identity

o 3.1 Related identities

4 Historic shorthands

5 Symmetry, shifts, and periodicity

o 5.1 Symmetry

o 5.2 Shifts and periodicity

6 Angle sum and difference identities

o 6.1 Matrix form

o 6.2 Sines and cosines of sums of infinitely many terms

o 6.3 Tangents of sums

o 6.4 Secants and cosecants of sums

7 Multiple-angle formulae

o 7.1 Double-angle, triple-angle, and half-angle formulae

o 7.2 Sine, cosine, and tangent of multiple angles

o 7.3 Chebyshev method

o 7.4 Tangent of an average

o 7.5 Vite's infinite product

8 Power-reduction formula

9 Product-to-sum and sum-to-product identitieso 9.1 Other related identities

o 9.2 Hermite's cotangent identity

o 9.3 Ptolemy's theorem

10 Linear combinations

11 Lagrange's trigonometric identities
http://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/Trigonometric_substitutionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Notationhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Notationhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Historic_shorthandshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Historic_shorthandshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetry.2C_shifts.2C_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetry.2C_shifts.2C_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetryhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetryhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Matrix_formhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Matrix_formhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sines_and_cosines_of_sums_of_infinitely_many_termshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sines_and_cosines_of_sums_of_infinitely_many_termshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangents_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangents_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Secants_and_cosecants_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Secants_and_cosecants_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle.2C_triple-angle.2C_and_half-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle.2C_triple-angle.2C_and_half-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sine.2C_cosine.2C_and_tangent_of_multiple_angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sine.2C_cosine.2C_and_tangent_of_multiple_angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Chebyshev_methodhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Chebyshev_methodhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_of_an_averagehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_of_an_averagehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Vi.C3.A8te.27s_infinite_producthttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Vi.C3.A8te.27s_infinite_producthttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulahttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulahttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Hermite.27s_cotangent_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Hermite.27s_cotangent_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Ptolemy.27s_theoremhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Ptolemy.27s_theoremhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange.27s_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange.27s_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Lagrange.27s_trigonometric_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Linear_combinationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Ptolemy.27s_theoremhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Hermite.27s_cotangent_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulahttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Vi.C3.A8te.27s_infinite_producthttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_of_an_averagehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Chebyshev_methodhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sine.2C_cosine.2C_and_tangent_of_multiple_angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle.2C_triple-angle.2C_and_half-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Multiple-angle_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Secants_and_cosecants_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangents_of_sumshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Sines_and_cosines_of_sums_of_infinitely_many_termshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Matrix_formhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Shifts_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetryhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Symmetry.2C_shifts.2C_and_periodicityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Historic_shorthandshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Related_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Pythagorean_identityhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angleshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Notationhttp://en.wikipedia.org/wiki/List_of_trigonometric_identitieshttp://en.wikipedia.org/wiki/Trigonometric_substitution


4/215

12 Other sums of trigonometric functions

13 Certain linear fractional transformations

14 Inverse trigonometric functions

o 14.1 Compositions of trig and inverse trig functions

15 Relation to the complex exponential function

16 Infinite product formulae

17 Identities without variables

o 17.1 Computing

o 17.2 A useful mnemonic for certain values of sines and cosines

o 17.3 Miscellany

o 17.4 An identity of Euclid

18 Composition of trigonometric functions

19 Calculus

o 19.1 Implications

20 Exponential definitions

21 Miscellaneous

o 21.1 Dirichlet kernel

o 21.2 Tangent half-angle substitution

22 See also

23 Notes

24 References

25 External links

Notation[edit source|editbeta]

Angles[edit source|editbeta]

This article usesGreek letterssuch asalpha(),beta(),gamma(), andtheta() to

representangles. Several differentunits of angle measureare widely used,

includingdegrees,radians, andgrads:

1 full circle = 360 degrees = 2 radians = 400 grads.

The following table shows the conversions for some common angles:

Degrees 30 60 120 150 210 240 300 330
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_sums_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_sums_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Certain_linear_fractional_transformationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Certain_linear_fractional_transformationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Compositions_of_trig_and_inverse_trig_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Compositions_of_trig_and_inverse_trig_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_the_complex_exponential_functionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_the_complex_exponential_functionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Infinite_product_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Infinite_product_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variableshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variableshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Computing_.CF.80http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Computing_.CF.80http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Computing_.CF.80http://en.wikipedia.org/wiki/List_of_trigonometric_identities#A_useful_mnemonic_for_certain_values_of_sines_and_cosineshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#A_useful_mnemonic_for_certain_values_of_sines_and_cosineshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellanyhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellanyhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#An_identity_of_Euclidhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#An_identity_of_Euclidhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Composition_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Composition_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Calculushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Calculushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Implicationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Implicationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellaneoushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellaneoushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Dirichlet_kernelhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Dirichlet_kernelhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_half-angle_substitutionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_half-angle_substitutionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#See_alsohttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#See_alsohttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Noteshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Noteshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Referenceshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Referenceshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#External_linkshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#External_linkshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=2http://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Angle#Unitshttp://en.wikipedia.org/wiki/Angle#Unitshttp://en.wikipedia.org/wiki/Angle#Unitshttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Angle#Unitshttp://en.wikipedia.org/wiki/Anglehttp://en.wikipedia.org/wiki/Thetahttp://en.wikipedia.org/wiki/Gammahttp://en.wikipedia.org/wiki/Beta_(letter)http://en.wikipedia.org/wiki/Alpha_(letter)http://en.wikipedia.org/wiki/Greek_lettershttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=2http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=1http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=1http://en.wikipedia.org/wiki/List_of_trigonometric_identities#External_linkshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Referenceshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Noteshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#See_alsohttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Tangent_half-angle_substitutionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Dirichlet_kernelhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellaneoushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Implicationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Calculushttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Composition_of_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#An_identity_of_Euclidhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Miscellanyhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#A_useful_mnemonic_for_certain_values_of_sines_and_cosineshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Computing_.CF.80http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Identities_without_variableshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Infinite_product_formulaehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Relation_to_the_complex_exponential_functionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Compositions_of_trig_and_inverse_trig_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Certain_linear_fractional_transformationshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Other_sums_of_trigonometric_functions


5/215

Radians

Grads 33 grad 66 grad 133 grad 166 grad 233 grad 266 grad 333 grad 366 grad

Degrees 45 90 135 180 225 270 315 360

Radians

Grads 50 grad 100 grad 150 grad 200 grad 250 grad 300 grad 350 grad 400 grad

Unless otherwise specified, all angles in this article are assumed to be in radians, but angles

ending in a degree symbol () are in degrees.

Trigonometric functions[edit source|editbeta]

The primary trigonometric functions are thesineandcosineof an angle. These are

sometimes abbreviated sin() and cos(), respectively, where is the angle, but the

parentheses around the angle are often omitted, e.g., sin and cos .

Thetangent(tan) of an angle is theratioof the sine to the cosine:

Finally, thereciprocal functionssecant (sec), cosecant (csc), and cotangent (cot) are the

reciprocals of the cosine, sine, and tangent:

These definitions are sometimes referred to asratio identities.

Inverse functions[edit source|editbeta]

Main article:Inverse trigonometric functions
http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=3http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Ratio_identitieshttp://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Ratio_identitieshttp://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Ratio_identitieshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=4http://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/wiki/Inverse_trigonometric_functionshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=4http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Ratio_identitieshttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Tangent_functionhttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=3http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=3http://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Degree_(angle)http://en.wikipedia.org/wiki/Grad_(angle)http://en.wikipedia.org/wiki/Radian


6/215

The inverse trigonometric functions are partialinverse functionsfor the trigonometric

functions. For example, the inverse function for the sine, known as the inverse

sine (sin1) orarcsine (arcsin or asin), satisfies

and

This article uses the notation below for inverse trigonometric functions:

Function sin cos tangent sec csc cot

Inverse arcsin arccos arctan arcsec arccsc arccot

Pythagorean identity[edit source|editbeta]

The basic relationship between the sine and the cosine is thePythagorean

trigonometric identity:

where cos2means (cos())

2and sin

2means (sin())2.

This can be viewed as a version of thePythagorean theorem, and

follows from the equation x2 + y2 = 1 for theunit circle. This equation

can be solved for either the sine or the cosine:

Related identities[edit source|editbeta]

Dividing the Pythagorean identity through by

eithercos2orsin2yields two other identities:

Using these identities together with the ratio identities, it is

possible to express any trigonometric function in terms of any

other(up toa plus or minus sign):

Each trigonometric function in terms of the other five.[1]
http://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=5http://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=6http://en.wikipedia.org/wiki/Up_tohttp://en.wikipedia.org/wiki/Up_tohttp://en.wikipedia.org/wiki/Up_tohttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-1http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-1http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-1http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-1http://en.wikipedia.org/wiki/Up_tohttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=6http://en.wikipedia.org/wiki/Unit_circlehttp://en.wikipedia.org/wiki/Pythagorean_theoremhttp://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/wiki/Pythagorean_trigonometric_identityhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=5http://en.wikipedia.org/wiki/Inverse_function


7/215

in

term

s of

Historic shorthands[edit source|editbeta]
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=7http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=7


8/215

All of the trigonometric functions of an angle can be constructed

geometrically in terms of a unit circle centered at O. Many of these terms are

no longer in common use.

Theversine, coversine, haversine, andexsecantwere used in

navigation. For example thehaversine formulawas used to

calculate the distance between two points on a sphere. They are

rarely used today.

Name(s) Abbreviation(s) Value[2]

versed sine,versine

versed cosine,vercosine

coversed sine,coversine

coversed cosine,covercosine
http://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Haversine_formulahttp://en.wikipedia.org/wiki/Haversine_formulahttp://en.wikipedia.org/wiki/Haversine_formulahttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2http://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/File:Circle-trig6.svghttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-2http://en.wikipedia.org/wiki/Haversine_formulahttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versine


9/215

half versed sine,haversine

half versed cosine,havercosine

half coversed sine,hacoversine

cohaversine

half coversed

cosine,hacovercosine

cohavercosine

exterior secant,exsecant

exterior cosecant,excosecant

chord

Symmetry, shifts, and periodicity[edit

source|editbeta]

By examining the unit circle, the following properties of the

trigonometric functions can be established.

Symmetry[edit source|editbeta]

When the trigonometric functions are reflected from certain

angles, the result is often one of the other trigonometric

functions. This leads to the following identities:

Reflected inReflected

in

Reflected in
http://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Chord_(geometry)http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=9http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=8http://en.wikipedia.org/wiki/Chord_(geometry)http://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Exsecanthttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versinehttp://en.wikipedia.org/wiki/Versine


10/215

[3] (co-function

identities)[4]

Shifts and periodicity[edit source|editbeta]

By shifting the function round by certain angles, it is often

possible to find different trigonometric functions that express

particular results more simply. Some examples of this are shown

by shifting functions round by /2, and 2 radians. Because

the periods of these functions are either or 2, there are

cases where the new function is exactly the same as the old

function without the shift.

Shift by /2Shift by

Period for tan and

cot[5]

Shift by 2Period for sin, cos, csc

and sec[6]

Angle sum and difference identities[edit

source|editbeta]
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=10http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-5http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-5http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-5http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-6http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-6http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-6http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=11http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-6http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-5http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=10http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=10http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-4http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-3


11/215

Illustration of angle addition formulae for the sine and cosine. Emphasized

segment is of unit length.

Illustration of the angle addition formula for the tangent. Emphasized segmentsare of unit length.

See also:#Product-to-sum and sum-to-product identities
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identitieshttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagramTangent.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/File:AngleAdditionDiagram.svghttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities


12/215

These are also known as the addition and subtraction

theorems orformulae. They were originally established by the

10th century Persian mathematicianAb al-Waf' Bzjn. One

method of proving these identities is to applyEuler's formula.

The use of the symbols and is described in the

articleplus-minus sign.

Sine [7][8]

Cosine [8][9]

Tangent[8][10]

Arcsine[11]

Arccosin

e [12]

Arctang

ent [13]

Matrix form[edit source|editbeta]

See also:matrix multiplication

The sum and difference formulae for sine and cosine can be

written inmatrixform as:
http://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%ABhttp://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%ABhttp://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%ABhttp://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%ABhttp://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%ABhttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-7http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-7http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-11http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-11http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-12http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-12http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-13http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=12http://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_(mathematics)http://en.wikipedia.org/wiki/Matrix_multiplicationhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=12http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=12http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-13http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-12http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-11http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_addition-8http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-7http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-7http://en.wikipedia.org/wiki/Plus-minus_signhttp://en.wikipedia.org/wiki/Euler%27s_formulahttp://en.wikipedia.org/wiki/Ab%C5%AB_al-Waf%C4%81%27_B%C5%ABzj%C4%81n%C4%AB


13/215

This shows that these matrices form arepresentationof the

rotation group in the plane (technically, thespecial

orthogonal groupSO(2)), since the composition law is

fulfilled: subsequent multiplications of a vector with these

two matrices yields the same result as the rotation by the

sum of the angles.

Sines and cosines of sums of infinitely

many terms[edit source|editbeta]

In these two identities an asymmetry appears that is

not seen in the case of sums of finitely many terms:

in each product, there are only finitely many sine

factors andcofinitelymany cosine factors.

If only finitely many of the terms iare nonzero, then

only finitely many of the terms on the right side willbe nonzero because sine factors will vanish, and in

each term, all but finitely many of the cosine factors

will be unity.

Tangents of sums[edit source|editbeta]
http://en.wikipedia.org/wiki/Group_representationhttp://en.wikipedia.org/wiki/Group_representationhttp://en.wikipedia.org/wiki/Group_representationhttp://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=13http://en.wikipedia.org/wiki/Cofinitenesshttp://en.wikipedia.org/wiki/Cofinitenesshttp://en.wikipedia.org/wiki/Cofinitenesshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=14http://en.wikipedia.org/wiki/Cofinitenesshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=13http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=13http://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/wiki/Special_orthogonal_grouphttp://en.wikipedia.org/wiki/Group_representation


14/215

Let ek(fork= 0, 1, 2, 3, ...) be the kth-

degreeelementary symmetric polynomialin the

variables

fori= 0, 1, 2, 3, ..., i.e.,

Then

The number of terms on the right side

depends on the number of terms on the

left side.

For example:
http://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomial


15/215

and so on. The case of only finitely

many terms can be proved

bymathematical induction.[14]

Secants and cosecants of

sums[edit source|editbeta]

where ek is the kth-

degreeelementary symmetric

polynomialin

the n variablesxi= tan i, i= 1, ..

., n, and the number of terms in

the denominator and the number

of factors in the product in the

numerator depend on the

number of terms in the sum on

the left. The case of only finitelymany terms can be proved by

mathematical induction on the

number of such terms. The

convergence of the series in the

denominators can be shown by

writing the secant identity in the

form

and then observing that the

left side converges if the

right side converges, and
http://en.wikipedia.org/wiki/Mathematical_inductionhttp://en.wikipedia.org/wiki/Mathematical_inductionhttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=15http://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/wiki/Elementary_symmetric_polynomialhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=15http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-14http://en.wikipedia.org/wiki/Mathematical_induction


16/215

similarly for the cosecant

identity.

For example,

Multiple-angleformulae[edit

source|editbeta]

Tn is

the nt

hChe

byshe

v

polyn

omial

[15]

Sn is

the nt

hspre

ad

polyn

omial

de

Moivr

e's

formu

la, i

s

thei

magin

ary

[16]
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=16http://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-16http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-16http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-16http://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/De_Moivre%27s_formulahttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/Spread_polynomialshttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/wiki/Chebyshev_polynomialshttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=16


17/215

unit

Double-angle,

triple-angle, and

half-angleformulae[edit

source|editbeta]

See also:Tangent half-

angle formula

These can be shown by

using either the sum and

difference identities or

the multiple-angle

formulae.

Double-angle

formulae[17][18]

Triple-angle formulae[15][19]

Half-angle formulae[20][21]
http://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/wiki/Imaginary_unithttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=17http://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-20http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-20http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-20http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-20http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-20http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-mathworld_multiple_angle-15http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-17http://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=17http://en.wikipedia.org/wiki/Imaginary_unit


18/215

The fact that the triple-

angle formula for sine

and cosine only involves

powers of a single

function allows one to

relate the geometric

problem of acompass

and straightedge

constructionofangle

trisectionto the algebraic

problem of solving

acubic equation, which
http://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Angle_trisectionhttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructionshttp://en.wikipedia.org/wiki/Compass_and_straightedge_constructions


19/215

allows one to prove that

this is in general

impossible using the

given tools, byfield

theory.

A formula for computing

the trigonometric

identities for the third-

angle exists, but it

requires finding the

zeroes of thecubic

equation

,

wherexis the value of

the sine function at some

angle and dis the known

value of the sine function

at the triple angle.

However,

thediscriminantof this

equation is negative, so

this equation has three

real roots (of which only

one is the solution within

the correct third-circle)

but none of these

solutions is reducible to

a real algebraic

expression, as they useintermediate complex

numbers under thecube

roots, (which may be

expressed in terms of

real-only functions only if
http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Discriminanthttp://en.wikipedia.org/wiki/Discriminanthttp://en.wikipedia.org/wiki/Discriminanthttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Cube_roothttp://en.wikipedia.org/wiki/Discriminanthttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Cubic_functionhttp://en.wikipedia.org/wiki/Field_(mathematics)http://en.wikipedia.org/wiki/Field_(mathematics)


20/215

using hyperbolic

functions).

Sine, cosine, and

tangent of

multiple

angles[edit

source|editbeta]

For specific multiples,

these follow from the

angle addition formulas,

while the general

formula was given by

16th century French

mathematicianVieta.

In each of these

two equations,

the first

parenthesized

term is

abinomial

coefficient, and

the final

trigonometric

function equals

one or minus

one or zero so

that half the

entries in each of

the sums are

removed.
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=18http://en.wikipedia.org/wiki/Franciscus_Vietahttp://en.wikipedia.org/wiki/Franciscus_Vietahttp://en.wikipedia.org/wiki/Franciscus_Vietahttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Binomial_coefficienthttp://en.wikipedia.org/wiki/Franciscus_Vietahttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=18


21/215

tan ncan be

written in terms

of tan using

the recurrence

relation:

cot ncan

be written in

terms of

cot using

the

recurrence

relation:

Cheby

shev

metho

d[edit

source|editbeta]

TheChe

byshev

method

is a

recursive

algorithm

for

finding

the nth m

ultiple

angle

formula

knowing
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=19http://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/wiki/Pafnuty_Chebyshevhttp://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=19


22/215

the

(n 1)th

and

(n 2)th f

ormulae.[

22]

The

cosine

fornxca

n be

compute

d from

the

cosine of

(n 1)x

and

(n 2)x

as

follows:

Simil

arly

sin(n

x)

can

be

com

pute

d

from

the

sines

of

(n

1)xa
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22http://en.wikipedia.org/wiki/List_of_trigonometric_identities#cite_note-22


23/215

nd

(n

2)x

F

o

r

t

h

e

t

a

n

g

e

n

t,

w

e

h

av

e

:

w

h

er

e

H

/

K


24/215

=

t

a

n

(

n

1

)x

.

T

a

n

g

e

n

t

o

f

a

n

av

e

r

a

g


25/215

e

[

e

d

i

t

s

o

u

r

c

e

|

e

d

i

t

b

e

t

a

]

S

e

t

t

i

n

g
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=20


26/215

e

i

t

h

e

r

o

r

t

o

0

g

i

v

e

s

t

h

e

us

u

a

l

t


27/215

a

n

g

e

n

t

h

a

l

f

-

an

g

l

e

f

o

r

m

u

l

.

V

i

te

'

s

i


28/215

n

f

i

n

it

e

p

r

o

d

uc

t

[

e

d

i

t

s

o

u

r

c

e

|

e

d

i

t

b

e
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=21


29/215

t

a

]
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&action=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=21


30/215
http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/index.php?title=List_of_trigonometric_identities&veaction=edit&section=22http://en.wikipedia.org/w/

Documents

Formula.doc