TRIGONOMETRIC FUNCTIONS OF

ANGLES

QUADRANTSThe coordinate axes divide the plane into four parts called quadrants. For any given angle in standard position, the measurement boundaries for each quadrant are summarized as follows:

x

y

o

Quadrant I

Quadrant III

Quadrant II

Quadrant IV

00 900 00 18090

00 270180 00 360270

( +, - )

( -, + ) ( +, + )

( -, - )

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE If is an angle in standard position, P(x, y) is any point other than the origin on the terminal side of , and , then

22 yxr

x

y

o

x

yr

)y,x(P

0y if ,y

rcsc

r

ysin

0x if ,x

rsec

r

xcos

0y if ,y

xcot

0x if ,x

ytan

x

y

o

sinAll Functionscsc

cosseccot

tan

SIGNS OF THE TRIGONOMETRIC FUNCTIONSEach of the trigonometric functions of an angle is given by two of the variables x, y and r associated with . Because r is always positive, the sign (+ or -) of a trigonometric function is determined by the signs of x and y, and therefore by the quadrant containing .

QUADRANTAL ANGLESAn angle in standard position whose terminal side lies on

the x or y-axis is called a quadrantal angle. The definitions of the trigonometric functions can be used to evaluate the trigonometric functions of the quadrantal angles 00, 900, 1800, 2700, and 3600 by using r equal to 1.

x

y

o1r

0y

1x

1r

1y

0x

1r

1y

0x

1r

0y

1x

REFERENCE ANGLE The reference angle of any angle is the positive angle formed by the terminal side of the angle and the nearest x-axis.

R

A summary of how to calculate the reference angle from a given angle is given below:

Quadrant I :Quadrant II :Quadrant III :Quadrant IV :

R

R

0R 180

0R 180

0R 360

EXAMPLE1. Determine the quadrant where the terminal side of

each angle lie when it is in standard position.

3

2 )a

0197 )b

2. The terminal side of angle in standard position passes through P. Draw and find the exact values of the six trigonometric functions of .

P(-3,-3) )a )3P(2,- )b

3. Determine the sign of the following trigonometric functions without the aid of calculator.

0301 csc )b0135 cot )a

EXAMPLE4. Find the exact values of the other five trigonometric

functions for an angle in standard position lying in the given quadrant.

5. Give the measure of the reference angle for each of the angle in standard position.

R

0110- )a 0505 )b

6. Find the exact values of the six trigonometric functions for each of the following angle without the aid of calculator.

3

5 )b

0135 )a

IV ,5

12 cot )a II ,2 sec)b