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7.4 Trigonometric Functions
of General Angles
In this section, we will study the following topics:
Evaluating
trig functions of any angle
Using
the unit circle to evaluate the trig functions of
quadrantal angles
Finding
Using
coterminal angles
reference angles to evaluate trig functions.
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Trig Functions of Any Angle
In 7.3, we looked at the definitions of the trig functions of acute angles
of a right triangle. In this section, we will expand upon those definitions
to include ANY angle.
We will be studying angles that are greater than 90° and less than 0°,
so we will need to consider the signs of the trig functions in each of the
quadrants.
We will start by looking at the definitions of the trig functions of any
angle.
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Definitions of Trig Functions of Any Angle
Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and
y
r
x
cos 
r
y
tan  
x
sin  
r  x2  y 2
r
y
r
sec 
x
x
cot  
y
y
csc 
(x, y)
r

x
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Example*
Let (-12, -5) be a point on the terminal side of . Find the exact
values of the six trig functions of .
y
First you must find the value of r:
r  x2  y2

-12
x
r
-5
(-12, -5)
4
Example (cont)
y

r
x
cos   
r
y
tan   
x
r
csc   
y
r
sec   
x
x
cot   
y
sin  
y

-12
(-12, -5)
x
13
-5
5
You Try!
Let (-3, 7) be a point on the terminal side of . Find the value
of the six trig functions of .
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The Signs of the Trig Functions
Since the radius is always positive (r > 0), the signs of
the trig functions are dependent upon the signs of x
and y.
Therefore, we can determine the sign of the functions
by knowing the quadrant in which the terminal side of
the angle lies.
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The Signs of the Trig Functions
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A trick to remember where each trig function is POSITIVE:
All Students Take Calculus
Translation:
A = All 3 functions are positive in Quad 1
S
A
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
T
C
C= Cosine function is positive in Quad 4
*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive,
but sine and cosine are negative; ...
**Reciprocal functions have the same sign. So cosecant is positive wherever sine
is positive, secant is positive wherever cosine is positive, …
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Example
Determine if the following functions are positive or negative:
sin 210°
cos 320°
cot (-135°)
csc 500°
tan 315°
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Example*
8
Given cos  
and cot 
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other trig function of .
 0, find the values of the five
Solution
First, determine the quadrant in which  lies. Since the cosine is negative
and the cotangent is positive, we know that  lies in Quadrant _____ .
x 8
cos  
r 17
Using the fact that x 2  y 2  r 2 , we can find y.
 -8
2
 y 2  17 
2
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Example* (cont)
Now we can find the values of the remaining trig functions:
x  8
y
sin   
r
x
cos   
r
y
tan   
x
y  15
r  17
r
csc   
y
r
sec   
x
x
cot   
y
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Another Example
3
Given cot    and 
8
other trig functions of .
   2 , find the values of the five
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Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal


3
side falls on one of the axes (...,  ,  , 0, ,  , , 2 ,...), we will
2
2
2
use the unit circle.

(0, 1) 2
Unit Circle:
(-1, 0)

Center (0, 0)

radius = 1

x2 + y2 = 1
(1, 0)

0
3
2
(0, -1)
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Now using the definitions of the trig functions with r = 1,
we have:
y y
r 1
sin     y
csc   
r 1
y y
x x
cos     x
r 1
y
tan  
x
r 1
sec   
x x
x
cot  
y
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Example*
Find the value of the six trig functions for   

(0, 1)
3
2
(-1, 0)
(1, 0)




2
(0, -1)
0

2
 
sin     y 
 2
 
cos     x 
 2
y
 
tan     
x
 2
1
 
csc     
y
 2
  1
sec     
x
 2
x
 
cot     
y
 2
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Example
Find the value of the six trig functions for
  7
sin  7   y 
cos  7   x 
y
tan  7   
x
1
csc  7   
y
1
sec  7   
x
x
cot  7   
y
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Coterminal Angles
Two angles
standard
are said coterminal to be if they
Sectionin4.1,
Figure position
4.4, Coterminal
Angles, pg. 248
have the same terminal sides.
 is a negative angle
coterminal to 
 is a positive angle (> 360°)
coterminal to 
In each of these illustrations, angles  and  are
coterminal.
Copyright © Houghton Mifflin Company. All rights reserved.
Digital Figures, 4–5
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Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by adding
or subtracting multiples of 360º or 2.
Example:
Find one positive and one negative angle that are coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º
Note: There are an infinite number of angles that are coterminal to 112 º.
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Example

Find one positive and one negative coterminal angle
 of
3
4
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Reference Angles
The values of the trig functions for non-acute angles (Quads II, III, IV)
can be found using the values of the corresponding reference angles.
I will use the notation  ' to represent an angle’s reference angle.
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Reference Angles
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Example
Find the reference angle for
Solution
  225
y
By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ', you would subtract 180°
from 225 °.
 '  225  180
 '  _____ 

'
x
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More Examples
Find the reference angles for the following angles.
1.
  210
2.  
5
4
3.   5.2
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So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.
For example,
sin 225  (sin 45)  
In Quad 3, sin is negative
1
2
45° is the ref angle
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Trig Functions of Common Angles
Using reference angles and the special reference triangles, we can
find the exact values of the common angles.
To find the value of a trig function for any common angle 
1.
Determine the quadrant in which the angle lies.
2.
Determine the reference angle.
3.
Use one of the special triangles to determine the function value
for the reference angle.
4.
Depending upon the quadrant in which  lies, use the
appropriate sign (+ or –).
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More Examples
Give the exact value of the trig function (without using a calculator).
1.
sin
5
6
2.
 3 
cos   
 4 
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More Examples
Give the exact value of the trig function (without using a calculator).
3. cot 660
4
4. csc
3
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Evaluating trig function of
“uncommon” angles
To find the value of the trig functions of angles that do NOT
reference 30°, 45°, or 60°, and are not quadrantal, we will use
the calculator. Round your answer to 4 decimal places, if
necessary.

Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree

To find the trig functions of csc, sec, and cot, use the reciprocal
identities.
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Examples
Evaluate the trig functions to four decimal places.
2. sec( 2.5)
3.
csc  23 38' 45"
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
End of Section 7.4
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