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 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.
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
• 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
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