Chapter 3

Trigonometric Functions of Angles

Section 3.3

Trigonometric Functions of Angles

P:(x,y)

Angles in Standard Position

Recall an angle in standard position is an angle that has its initial side on the positive x-axis. We can use any point on the angles terminal side to find the values of the trigonometric ratios. If the coordinates of the point P are (x,y) and the distance the point P is from the origin is r we get the following values for the trigonometric ratios.

x-axis

y-axis

x

y

22 yxr

22sin

yx

y

r

y

22cos

yx

x

r

x

x

ytan

y

xcot

x

yx

x

r 22

sec

y

yx

y

r 22

csc

In the example to the right with the coordinates of P at the point (1,3)

1 2

123

P:(1,3)

10

31 22

r

rr10

3sin

10

1cos

1

3tan

3

1cot

1

10sec

3

10csc

It is important to realize that it does not matter what point you select on the terminal side of the angle the trigonometric ratios will be the same because the triangles are similar. The triangle with its vertex at P1 is similar to the triangle with its vertex at P2 and the length of the sides are proportional (equal ratios).

P1

P2

Signs of Trigonometric Functions

The trigonometric ratios now are defined no matter where the terminal side of the angle is. It can be in any if the four quadrants. Since the values for the xy-coordinates are different signs (±) depending on the quadrant the trigonometric ratios will be also. The value for r is always positive. The chart below shows the signs of the trigonometric ratios.

x pos (+)y pos (+)

x neg (-)y pos (+)

x pos (+)y neg (-)

x neg (-)y neg (-)

Quadrant sin cos tan cot sec csc

I + + + + + +

II + - - - - +

III - - + + - -

IV - + - - + -

Find the values of the six trigonometric functions if the point (-3,2) is on the terminal side of the angle. We find the value for r (distance from the origin) first.

-3

2r

13492)3( 22 r

13

2sin

13

3cos

3

2tan

3

13sec

2

3cot

2

13csc

Reference Angles

The reference angle for an angle is the angle made when you drop a line straight down to the x-axis. it is the angle made by the x-axis regardless of what side of it you are on.

120

60

225

45

330

30-300

60

Reference angles are useful to help you find the values for trigonometric functions for many angles of the circle. For example if we want to find the trigonometric ratios for 150. We know the reference angle is 30 and we for a 30-60-90 triangle. The sides are in the ratios we mentioned before.

150

30

2

3

2

11

2

1150sin

2

3150cos

3

1150tan

3150cot

3

2150sec

2150csc

Identities

sin

coscot

cos

sintan

cot

1tan

cos

1sec

sin

1csc

222222 csccot1sec1tan1cossin