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Chapter 5 Trigonometric Equations. 5.4. Trigonometric Identities. 5.4. 1. MATHPOWER TM 12, WESTERN EDITION. Trigonometric Identities. A trigonometric equation is an equation that involves at least one trigonometric function of a variable. The - PowerPoint PPT Presentation
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MATHPOWERTM 12, WESTERN EDITION
5.4
5.4.1
Chapter 5 Trigonometric Equations
A trigonometric equation is an equation that involvesat least one trigonometric function of a variable. Theequation is a trigonometric identity if it is true for allvalues of the variable for which both sides of theequation are defined.
Trigonometric Identities
Prove that tan sincos
.
5.4.2
Recall the basictrig identities:
sin y
r
cos x
r
tan y
x
5.4.3
Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
sinx x sinx =
cos 1
cos
cos 2cos
1
cos
cos 2 1
cos
sin A cos A 2 sin2 A 2sin Acos A cos 2 A
Trigonometric Identities [cont’d]
5.4.4
Identities can be used to simplify trigonometric expressions.
Simplifying Trigonometric Expressions
cos sin tana)
Simplify.
b)cot2
1 sin2
5.4.5
5.4.6
Simplifying Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x d)csc x
tan x cot x
5.4.7
Proving an Identity
Steps in Proving Identities
1. Start with the more complex side of the identity and work with it exclusively to transform the expression into the simpler side of the identity.2. Look for algebraic simplifications:
3. Look for trigonometric simplifications:
4. Keep the simpler side of the identity in mind.
5.4.8
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
1 + sec x
b) sec x = tan x csc x
sec x
c) tan x sin x + cos x = sec x
sec x
d) sin4x - cos4x = 1 - 2cos2 x
1 - 2cos2x
e)
1
1 cos x
1
1 cosx 2 csc2 x
2 csc2 x
Proving an Identity
5.4.9
Proving an Identity
5.4.10
f)
cos A
1 sin A
1 sin A
cos A 2 sec A
Using Exact Values to Prove an Identity
5.4.11
Consider sin x
1 cos x
1 cosx
sinx.
b) Verify that this statement is true for x = 6
.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.
a)
sin x
1 cos x
1 cosx
sinx
b) Verify that this statement is true for x = 6
.
Rationalize thedenominator:
Using Exact Values to Prove an Identity [cont’d]
5.4.12
c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.
Using Exact Values to Prove an Identity [cont’d]
5.4.13
sin x
1 cos x
1 cosx
sinx
1 cos x
sin x
Note the left side of theequation has the restriction
Therefore, where n is any integer.
The right side of theequation has the restriction
Therefore, And , where n is any integer.
Restrictions:
Proving an Equation is an Identity
Consider the equationsin2 A 1
sin2 A sin A1
1
sin A.
b) Verify that this statement is true for x = 2.4 rad.
a) Use a graph to verify that the equation is an identity.
c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.
a)
5.4.14
b) Verify that this statement is true for x = 2.4 rad.
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2 A sin A 1
1
sin A
5.4.15
5.4.16
Proving an Equation is an Identity [cont’d]
sin2 A 1
sin2 A sin A 1
1
sin A
c) Use an algebraic approach to prove that the identity is true in general. State any restrictions.
1 1
sin A
L.S. = R.S.
Note the left side of theequation has the restriction: sin2A - sin A ≠ 0
A 0, or A
2
Therefore, A 0 2 n or
A +2n, or
A 2
2 n, where n is
any integer.
The right side of theequation has the restrictionsin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0, + 2n,where n is any integer.
sin A(sin A - 1) ≠ 0
5.4.16
Suggested Questions:Pages 264 and 265A 1-10, 21-25, 37, 11, 13, 16B 12, 20, 26-34