4.4 Trigonmetric functions of Any Angle

Objective

• Evaluate trigonometric functions of any angle

• Use reference angles to evaluate trig functions

Definitions of Trigonometric Functions of any Angle

• Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and

2 2 0r x y

sin cos

tan , 0 cot 0,

sec , 0 csc , 0

y x

r ry xx y

x y

r rx y

x y

• The cosecant function is the reciprocal of the sine.

• The secant function is the reciprocal of the cosine.

• The cotangent function is the reciprocal of the tangent function.

Example 1

• Let (-3, 4) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ.

2 2

2 2( 3) 4

9 16

5

4sin

53

cos54

tan3

r x y

r

r

r

y

rx

ry

x

Example 2

• Let (2, 5) be a point on the terminal side of θ. Find the sine, cosine, and tangent of θ.

2 2

2 2(2) 5

4 25

29

5 5 29 5 29sin

2929 29 29

4 4 29 4 29cos

2929 29 295

tan4

r x y

r

r

r

y

r

x

r

y

x

Signs of the Trigonometric Functions

Signs of the Trig Functions

A means that all trig. functions are positive.S means that all sine and cosecant functions are positive.T means that all tangent and cotangent functions are positive.C means that all cosine and secant functions are positive.

Example 3

• State whether each value is positive, negative, or zero.

• a) cos 75° positive

• b) sin 3π 0

• c) cos 5π negative

• d) sin(-3π) 0

Example 4

• Given.

2 2

2 2

2

2

4sin tan 0, cos csc .

54

sin , implies y = 4 and r = 55

since tan <0, and y = 4, is in the II quadrant

5 4

25 16

9

3, since is in II, x = -3

3 5cos , csc

5 4

and find and

y

r

r x y

x

x

x

x

x r

r y

Example 5

• Angle θ is in standard position with its terminal side in the third quadrant. Find the exact value of cos θ if

2 2

2 2

2

2

1sin

21

sin ,implies y = -1, r = 22

2 ( 1)

4 1

3 , 3,since is in III, 3

3cos

2

y

r

r x y

x

x

x x x

x

r

Example 6

• Angle θ is in standard position with its terminal side in the fourth quadrant. Find the exact value of sin θ if

2 2

2 2

2

2

4cos

74

cos ,implies x = 4, r = 77

7 4

49 16

33 , 33,since is in IV, y 33

33sin

7

x

r

r x y

y

y

y y

y

r

Reference Angles

• Definition

• Let θ be an angle in standard position. Its reference angle is the acute angle θ’ formed by the terminal side of θ and the horizontal axis.

Reference angles

Example 7

• Finding reference angles.

. 213

. 1.7

. 144

a

b

c

Trigonometric Values of Common Angles

Example 8

• Use the reference angle to find sin θ, cos θ, and tan θ for each value of

. 150 is in II so ' 180 150 30

1 3 1 3sin 30 ,cos30 implies sin150 ,cos ,

2 2 2 2

1 31/ 2tan1503 / 2 33

. 330 is in IV so ' 360 330 30

1 3 3sin 330 ,cos330 , tan 330

2 2 37 7

. is in III, so '6 6 6

a

b

c

30

7 1 7 3 7 3sin ,cos , tan

6 2 6 2 6 3

Example 9

• Determine the values of θ for which

0 2 ,For

1sin , looking at the unit circle

25

,6 6

• If the value of one of the trig functions of any angle is known, a calculator can be used to determine the angles having that value.

Example 10

• Find values of θ, where • to the nearest tenth of a degree.

0 360

. cos .9266

Make sure calculator is in degrees

2nd cos(.9266) = 22.1

. sin 0.6009

2nd sin(-0.6009) 36.9

. tan .2309

2nd tan( .2309) 13

a

b

c

Example 11

• Find values of θ, where• To the nearest hundredth of a radian.

0 2

. tan 3.009

Make sure calculator is in radians

2nd tan(3.009) 1.25

. cot 4.69

2nd tan( 1/ 4.69) .21

. sec 8.2986

3nd cos(1/8.2986) 1.44

a

radians

b

radians

c

radians