Trigonometric functions

MEASUREMENT OF ANGLES

1. Angle: The measurement of an angle is the amount of rotation from the initial side to the terminal side.

2. Sense of an Angle: The sense of an angle is +ve or –ve based on whether the initial side rotates in the anti- clock-wise or clockwise direction to get the terminal side.

System of measuring angles

There are three systems of measuring angles such as

(i) Sexagesimal system

(ii) Centisimal system

(iii) Circular system

4. In sexagesimal system, we have 1 right angle = 90°

1° = 60'

1' = 60"

5. In centasimal system, we have 1 right angle = 100g

1g= 100'

1' = 100"

6. In circular system, the unit of measurement is radian

Radian: One radian is the measure of an angle sub- tended at the centre of a circle by an arc of length equal to the radius of the circle.

Notes

(i) When an angle is expressed in radians, the word ra- dian is omitted.

(ii) Since 180° = pi radian = 22/(7×180)radian =0.01746 radian

(iii) 1 radian = 180°/(pi) = 180×7/(22)=57°16'22"

(iv) The angle between two consecutive digits is 30°(pi/6)radians

(v) The hour hand rotates through an angle of 30° in one hour (i.e. 1/2 in one minute)

(vi) The minute hand rotates through an angle of 6° in one minute.

(vii) The relation amongst three systems of measurement of an angle is D/90°=G/100=2R/pi

(viii) The number of radians in an angle subtended by an arc of a circle at the centre is Arc/Radius i.e. θ=s/r

Trigonometric identities An equation invloving trigonometric functions which is true for all those angles for which the functions are defined is called a trigonometric identities. Fundamental trigonometric identities

SIGNS OF THE TRIGONOMETRIC RATIOS OR FUNCTIONS

as Above, we have introduced six trigonometric ratios. Their signs depend on the quadrant in which the terminal side of the angle lies.t

I Quadrant- all are +vet

II Quadrant- sin, cosec are +ve and rest are -vet

III Quadrant- tan, cot are +ve a d rest are - vet

IV Quadrant- cos, sec are +ve and rest are -ve t

"SIMPLE RULE TO REMEMBER SIGN OF TRIGONOMETRIC FUNCTIONS: A crude aid to memorise the signs of trigonometrical ratios in different quadrants is the four word phrase "ALL SCHOOL TO COLLEGE". The first letter of the first word in this phrase is 'A'. This may be taken to indicate that all trigonometric ratios are positive in the first quadrant.The first letter of the second word is 'S'. This indicates that sine and its reciprocal are positive in the second quadrant. The first letter of third word is 'T'. This may be taken as to indicate that tangent and its reciprocal are positive in the third quadrant. The first letter of the fourth word in the phrase is 'C' which may be taken as to indicate that only cosine and its reciprocal are positive in the fourth quadrant."