INTRODUCTIOINTRODUCTIONN

THE WORD TRIGONOMETRY IS DERIVED FROM THE GREEK WORDS TRIGON AND METRON AND IT MEANS MEASURING THE SIDES OF A TRIANGLE.

THE SUBJECT WAS ORIGINALLY DEVELOPED TO SOLVE GEOMETRIC PROBLEMS INVOLVING TRIANGLES.

CURRENTLY TRIGONOMETRY IS USED IN MANY AREAS SUCH AS DESIGNING ELECTRICAL CIRCUITS ,DESCRIBING THE STATE OF AN ATOM,PREDICTING THE HEIGHTS OF TIDES IN OCEAN AND IN MANY OTHER AREAS.

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KEY FOR TRIGONOMETRIC KEY FOR TRIGONOMETRIC

FUNCTIONSFUNCTIONS

INTRODUCTIONINTRODUCTION

ANGLESANGLES

DEGREE MEASUREDEGREE MEASURE

RADIAN RADIAN

FORMULASFORMULAS

AnAngglesles Measuring angles

Units

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Measuring AngleMeasuring Angle

The value of The value of θθ thus defined is independent of the size thus defined is independent of the size of the circle: if the length of the radius is changed of the circle: if the length of the radius is changed then the arc length changes in the same proportion, then the arc length changes in the same proportion, so the ratio so the ratio ss//rr is unaltered. is unaltered.

In many geometrical situations, angles that differ by In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a equivalent (it makes no difference how many times a line is rotated through a full circle because it always line is rotated through a full circle because it always ends up in the same place). However, this is not ends up in the same place). However, this is not always the case. For example, when tracing a curve always the case. For example, when tracing a curve such as a such as a spiralspiral using using polar coordinatespolar coordinates, an extra full , an extra full turn gives rise to a quite different point on the curve.turn gives rise to a quite different point on the curve.

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Degrees 30° 45° 60° 90°

Radians

Degrees 120° 150° 180° 360°

Radians

UnitsUnits Angles are considered dimensionless, since they are Angles are considered dimensionless, since they are

defined as the ratio of lengths. defined as the ratio of lengths.

With the notable exception of the radian, most units of angular With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one measurement are defined such that one full circle (i.e. one revolution) is equal to revolution) is equal to nn units, for some whole number units, for some whole number nn (for (for example, in the case of degrees, example, in the case of degrees, nn = 360). This is equivalent to = 360). This is equivalent to setting setting kk = = nn/2/2ππ in the formula above. (To see why, note that one in the formula above. (To see why, note that one full circle corresponds to an arc equal in length to the circle's full circle corresponds to an arc equal in length to the circle's circumferencecircumference, which is 2, which is 2πrπr, so , so ss = 2 = 2πrπr. Substituting, we get . Substituting, we get θθ = = ksks//rr = 2 = 2πkπk. But if one complete circle is to have a numerical . But if one complete circle is to have a numerical angular value of angular value of nn, then we need , then we need θθ = = nn. This is achieved by . This is achieved by setting setting kk = = nn/2/2ππ.).)

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Degree MeasureDegree Measure• The The degreedegree, denoted by a small superscript , denoted by a small superscript

circle (°) is 1/360 of a full circle, so one full circle circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old is 360°. One advantage of this old sexagesimalsexagesimal subunit is that many angles common in simple subunit is that many angles common in simple geometry are measured as a whole number of geometry are measured as a whole number of degrees. (The problem of having degrees. (The problem of having allall "interesting" angles measured as whole "interesting" angles measured as whole numbers is of course insolvable.) Fractions of a numbers is of course insolvable.) Fractions of a degree may be written in normal decimal degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), notation (e.g. 3.5° for three and a half degrees), but the following sexagesimal subunits of the but the following sexagesimal subunits of the "degree-minute-second" system are also in use, "degree-minute-second" system are also in use, especially for especially for geographical coordinatesgeographical coordinates and in and in astronomyastronomy and and ballisticsballistics: :

θθ = = ss//rr rad = 1 rad. rad = 1 rad.HOMEAngles

Radian • The radian is the angle subtended by an arc of a

circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system

• In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

• The value of thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

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Trigonometric functionsThe tangent (tan) of an angle is the ratio of the sine to the cosine:

Pythagorean identity

The basic relationship between the sine and the cosine is the Pythagorean trigonometric identity:

• sin(π/2–x) =cos xsin(π/2–x) =cos x• sin(sin(ππ/2+x) =cos x/2+x) =cos x• cos(π/2–x) =sin x cos(π/2–x) =sin x • cos(cos(ππ/2+x) =-sin x/2+x) =-sin x • cos(cos(ππ/2-x) =cos x/2-x) =cos x• sin(sin(ππ/2-x) =-sin x/2-x) =-sin x

Sign of trigonometric function

• sin (x + y) =sin x cos y + cos x sin ysin (x + y) =sin x cos y + cos x sin y• sin (x – y) =sin x cos y – cos x sin ysin (x – y) =sin x cos y – cos x sin y• cos (x + y) =cos x cos y – sin x sin ycos (x + y) =cos x cos y – sin x sin y• cos (x – y) =cos x cos y + sin x sin ycos (x – y) =cos x cos y + sin x sin y

• Tan(x+y) = tanx +tany/1- tanx tany • Tan (x-y) = tanx-tany/1+tanx tany• Cot (x+y) = cotx+coty/1-cotx coty • Cot (x-y) = cotx-coty/1+cotx coty• cos2x= = cos

Multiple-angle formulae

• Made By:-Made By:- Anand Anand YadavYadav Class-Class-XI XI BB

• Made By:-Made By:- Anand Anand YadavYadav Class-Class-XI XI BB