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4.4 Trig Functions of Any Angle
Objectives:
Evaluate trigonometric functions of any angle
Use reference angles to evaluate trig functions
Trig Functions of Any Angle
Given an angle  in standard position with
(x,y) a point on the terminal side of 


Precalculus
y
sin  
r
r
csc   , y  0
y
y
x
cos 
r
r
sec   , x  0
x
(x,y)

y
tan   , x  0
x

x
cot   , y  0
y
4.4 Trigonometric Functions of Any Angle
r

x
r  x2  y2
2
Example 1
· Let (-3,4) be a point on the terminal side
of . Find the sine, cosine and tangent of
.
Precalculus
4.4 Trigonometric Functions of Any Angle
3
Example 2
5
· Given tan    and cos  0
4
· Find sin  and sec 

Precalculus
4.4 Trigonometric Functions of Any Angle
4
Reference Angles
· Let  be an angle in standard position. Its
reference angle is the acute angle ’
formed by the terminal side of  and the
x-axis.
’


’
Precalculus
4.4 Trigonometric Functions of Any Angle

’
5
Calculating Reference Angles
’



’
 '    (radians)
 '180   (degrees)


Precalculus
’
'    (radians)
'  180 (degrees)
 ' 2   (radians)
 ' 360   (degrees)

4.4 Trigonometric Functions of Any Angle
6
Example 3
· Find the reference angle ’
·  = 300°
·  = 2.3
·  = -135°
Precalculus
4.4 Trigonometric Functions of Any Angle
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Evaluating Trig Functions of
Any Angle
To find the value of a trig function of any
angle :
1. Evaluate the function for the associated
reference angle ’.
2. Determine the sign of the trig function
based on the quadrant in which  lies.
Precalculus
4.4 Trigonometric Functions of Any Angle
8

 Precalculus
Example 4
· Use reference angles to evaluate the trig
function
4
· cos
3
· tan210
11
· csc
4
4.4 Trigonometric Functions of Any Angle
9
Example 5
· Let  be an angle in Quadrant II such that
sin =1/3. Find cos  and tan  by using
trig identities.
Precalculus
4.4 Trigonometric Functions of Any Angle
10
Closure
· Explain how to use reference angles to
determine trig functions of any angle
Precalculus
4.4 Trigonometric Functions of Any Angle
11
Assignment
· 4.4 page 320: 3-102 every 3rd, 111, 112
Precalculus
4.4 Trigonometric Functions of Any Angle
12
Bellwork
1. Find sin 

4
7
2. Determine the sign (positive or
negative) of the six trig functions for
angles in Quadrant I, Quadrant II,
Quadrant III, and Quadrant IV
Precalculus
4.4 Trigonometric Functions of Any Angle
13