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Chapter 4
Trigonometric
Functions
4.4 Trigonometric
Functions of Any Angle
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
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Objectives:
Use the definitions of trigonometric functions of any
angle.
Use the signs of the trigonometric functions.
Find reference angles.
Use reference angles to evaluate trigonometric
functions.
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Definitions of Trigonometric Functions of Any Angle
Let  be any angle in standard position and let P = (x, y)
be a point on the terminal side of  If
r  x 2  y 2 is
the distance from (0, 0) to (x, y), the six trigonometric
functions of  are defined by the following ratios:
y
sin  
r
r
csc  , y  0
y
x
cos 
r
y
tan   , x  0
x
r
sec  , x  0
x
x
cot   , y  0
y
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Example: Evaluating Trigonometric Functions
Let P = (1, –3) be a point on the terminal side of  Find
each of the six trigonometric functions of 
P = (1, –3) is a point on the terminal side of 
x = 1 and y = –3
r  x 2  y 2  (1)2  (3)2  1  9  10
y
3
3 10
3 10
sin   


r
10
10
10 10
1
1
10
10
x


cos  
r
10 10 10
10
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Example: Evaluating Trigonometric Functions
(continued)
Let P = (1, –3) be a point on the terminal side of  Find
each of the six trigonometric functions of 
We have found that r  10.
y 3
 3
tan   
x 1
x
r
10
1
cot    
csc   
y
y
3
3
10
r
 10
sec  
1
x
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Example: Evaluating Trigonometric Functions
(continued)
Let P = (1, –3) be a point on the terminal side of  Find
each of the six trigonometric functions of 
3 10
sin   
10
10
csc  
3
10
cos 
10
sec  10
tan   3
1
cot   
3
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Example: Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the cosine function and the
cosecant function at the following quadrantal angle:
  0  0
If   0  0 radians, then the terminal side of the
angle is on the positive x-axis. Let us select the point
P = (1, 0) with x = 1 and y = 0.
x 1
cos    1
r 1
r
1
csc  
csc is undefined.
y 0
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Example: Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the cosine function and the cosecant
function at the following quadrantal angle:

  90 
If   90  radians, then the terminal side of the
2
angle is on the positive y-axis. Let us select the point

2
P = (0, 1) with x = 0 and y = 1.
x 0
cos    0
r 1
r 1
csc    1
y 1
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Example: Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the cosine function and the cosecant
function at the following quadrantal angle:   180  
If   180   radians, then the terminal side of the
angle is on the positive x-axis. Let us select the point
P = (–1, 0) with x = –1 and y = 0.
x 1
cos    1
r 1
r
1
csc  
y 0
csc is undefined.
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Example: Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the cosine function and the cosecant
3
function at the following quadrantal angle:   270 
2
3
If   270 
radians, then the terminal side of the
2
angle is on the negative y-axis. Let us select the point
P = (0, –1) with x = 0 and y = –1.
x 0
cos    0
r 1
r 1
csc    1
y 1
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The Signs of the Trigonometric Functions
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Example: Finding the Quadrant in Which an Angle Lies
If sin   and cos  0, name the quadrant in which
the angle  lies.
 lies in Quadrant III.
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Example: Evaluating Trigonometric Functions
1
Given tan    and cos  0, find sin  and sec .
3
Because both the tangent and the cosine are negative, 
lies in Quadrant II.
y
1
x  3, y  1
tan   
x 3
r  x 2  y 2  (3)2  (1)2  9  1  10
y
1
10
10
sin   

r
10 10 10
10
10
r

sec  
3
x 3
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Definition of a Reference Angle
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Example: Finding Reference Angles
Find the reference angle,   for each of the following
angles:
a.   210      180  210  180  30
7
b.  
4
7 8 7 
   2    2 



4
4 4
4
c.   240    60
d.   3.6
       3.6  3.14  0.46
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Finding Reference Angles for Angles Greater Than 360°
(2 ) or Less Than –360°( 2 )
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Example: Finding Reference Angles
Find the reference angle for each of the following angles:
a.   665
   360  305  55
15
b.  
4
11
c.   
3
7 8 7 
   2 



4
4
4 4
11 12 
  


3
3
3
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Using Reference Angles to Evaluate Trigonometric
Functions
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A Procedure for Using Reference Angles to Evaluate
Trigonometric Functions
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Example: Using Reference Angles to Evaluate
Trigonometric Functions
Use reference angles to find the exact value of sin135.
Step 1 Find the reference angle,   and sin 
   360    360  300  60
Step 2 Use the quadrant in which  lies to prefix the
appropriate sign to the function value in step 1.
3
sin 300   sin 60  
2
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Example: Using Reference Angles to Evaluate
Trigonometric Functions
5
Use reference angles to find the exact value of tan .
4
Step 1 Find the reference angle,   and tan 
5 4 
     

4
4 4
Step 2 Use the quadrant in which  lies to prefix the
appropriate sign to the function value in step 1.
5

tan
 tan  1
4
4
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Example: Using Reference Angles to Evaluate
Trigonometric Functions


Use reference angles to find the exact value of sec    .
 6
Step 1 Find the reference angle,   and sec .
 12 

    2   

6
6
6
Step 2 Use the quadrant in which  lies to prefix the
appropriate sign to the function value in step 1.

 2 3

sec     sec 
6
 6
3
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