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Sect. 4.4
Trig Functions:
At Any Angle
Trig Functions of Any Angle
In 4.3, 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.
Definitions of Trig Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on
the terminal side of  and
r 2  x2  y 2
y
sin  
r
x
cos 
r
y
tan  
x
or
r  x2  y 2
r
csc 
y
r
sec 
x
x
cot  
y
y
(x, y)
r

x
Ex 1 in textbook:
Let (–3, 4) be a point on the terminal side of  .
Find the value of the six trig functions of  .
Solution:
x = -3, y = 4, still need r
csc 
r 5

y 4
r
5
sec    
x
3
cot  
x
3

y
4
Let’s do one together
Let (-12, -5) be a point on the terminal side of . Find the
exact values of the six trig functions of .
First we must find the value of r :
r  x2  y2
y

x
-12
r
13
(-12, -5)
-5
y

r
x
cos   
r
y
tan   
x
r
csc   
y
r
sec   
x
x
cot   
y
sin  
You Try…
Let (-3, 7) be a point on the terminal side of . Find the
value of the six trig functions of .
Evaluating Trig Functions
When Given a Point
To find the value of a trig function when given a point…
1.
Calculate r using x2 + y2 = r2.
2. Use the definition of the trig functions to write the ratios
using the appropriate x, y and r values.
3. Depending upon the quadrant in which  lies, use the
appropriate sign (+ or –).
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.
Signs of Trig Functions
A trick to remember where each trig function is POSITIVE:
All Students Take Calculus
Translation:
S
T
A
C
A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4
**Reciprocal functions have the same sign. So cosecant is positive wherever
sine is positive, secant is positive wherever cosine is positive, …
Ex 2 in textbook:
2
Given sin    and tan   0 , find cos
3
and cot  .
Solution:
Note  lies in Quad III because that is the only
quad in which sine is – and tangent is +. By letting
y
2
sin     , you can let y = -2 and r = 3.
r
3
x  5
r 2  x2  y 2
cos   
r
3
2
2
2
3  x  (2)
94  x
5x
x  5
cot   
y
2
Let’s do one together
8
and cot   0 , find the values of the five
17
other trig function of .
Given cos  
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  y 2  17 
2
x  8
2
y  15
r  17
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
You Try…
3
Given cot    and
8
    2 , find the values of the
five other trig functions of .
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.
Let  be an angle in standard position. Its reference angle is
the acute angle  ‘ formed by the terminal side of  and the
horizontal axis.
Example (like Ex 4 in TB)
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 °.

'
x
 '  225  180
 '  _____ 
Let’s do one together…
Find the reference angle for   325o.
Solution
To find  ', you would add
-325° to 360°.
 '  360   325
 '  35
You Try…
Find the reference angles for the following angles.
1.
  210
2.  
5
4
3.   5.2
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, find the exact value of sin 225 .
2
sin 225  (sin 45)  
2
In Quad 3, sine
is negative
45° is the ref
angle
Let’s do one together…
Give the exact value of the trig function
(without using a calculator).
1. cot 660
14
2. csc 
3
You Try…
Evaluate each trigonometric function.
1. cos
17
6
2.
tan  210o 
Evaluate Trig Functions Using
Reference 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 by subtracting from the
value of the horizontal axis that it is closest to.
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 –).
Closure
Explain how to find the 6 trig functions of
any angle. You may use the following as
an example.
y

x
22