Finding Exact Values For
Trigonometry Functions (Then
Using those Values to Evaluate
Trigonometry functions and Solve
Trigonometry Equations)
Review: Special Right Triangles
Find the exact values
of the missing side
lengths:
π60°
/3
1
30°-60°-90°
The “short leg” is
half the
hypotenuse
1
2
π45°
/4
1
π30°
/6
The “long
leg” is the
short leg
multiplied by
√3
3
2
π /°4
45
The hypotenuse is any
leg multiplied by √2
45°-45°-90°
OR
Isosceles Right
2
2
2
2
Exact Coordinates on the Unit Circle
The angles
The angles
0,1


1
that have the
from the
π90°
/2
same
special right
2π / 3
π60°
/3
120°
3π / 4
π45°
/4
reference
135°
triangles
angles as the
have exact
5π / 6
π 30°
/6
150°
angles
coordinates
from  1, 0  180°
π
0 1,0 
0°
special
-1
1
right
The x and
11π
/6
triangles
7π / 6
330°
210°
have exact
7π315°
/4
y-intercepts
5π / 4
225°
5π
/3
coordinates
4π
/3
300°
240°
obviously
3π
/2
270°
-1  0, 1
have exact
coordinates
Exact Coordinates on the Unit Circle
 1 3
  ,

 2 2 

2 2
,
 

2
2 


3 1
, 
 
 2 2
1  0,1
π/2
2π / 3
3π / 4
These
5π / 6
coordinates
tell you  1, 0  π
the exact
-1
values of

3 1


cosine and  2 ,  2  7π / 6
sine for 16
5π / 4


2
2
,
 

angles.
4π / 3
2 
 2
 1
3
  , 

2 
 2
1 3
 ,

2 2 
π/3
π/4
1
1
 2 2
,


 2 2 

1
45°
60°
30°
1/2 2 3
0
2 2
11π / 6
7π / 4


5π / 3

3π / 2 

-1  0, 1

3 1
3π / 6
, 

2
 2 2
2
2 1/2
1
3
 , 

2 
2
1,0
1
 3 1
,  

2
 2
2
2
,

2
2 
They need
to be
memorized.
Exact Coordinates on the Unit Circle
 1 3
  ,

 2 2 

2 2
,
 

2
2 

120°

3 1
, 
 
 2 2
1  0,1
90°
 1
3
  , 

2 
 2
 2 2
,


 2 2 
60°
135°
These
150°
coordinates
tell you  1, 0  180°
the exact
-1
values of

3 1


cosine and  2 ,  2  210°
sine for 16
225°

2
2
,
 

angles.
240°
2 
 2
1 3
 ,

2 2 
45°
 3 1
, 

 2 2
30°
0°
330°
315°
270°
1,0
1
 3 1
,  

2
 2
 2
2
,



300°
2 
 2
1
3
 , 

2 
2
-1  0, 1
They need
to be
memorized.
NOTE
The coordinates on that graph tell you the
exact values of cosine and sine for 16
angles. They need to be memorized for all
of the included angles.
If you do not wish to memorize the unit circle
or use special right triangles, the following
is a trick to assist in memorization.
Reference Angle
On the left are 3 reference angles that we know exact trig values
for. To find the reference angle for angles not in the 1st quadrant
(the angles at right), ignore the integer in the numerator.

6

4

3
: 30
: 45
: 60
5 7 11
,
,
6 6 6
3 5 7
, ,
4 4 4
2 4 5
,
,
3 3 3
NOTE:
Multiply the
number in
the
numerator
by the
degree to
find the
angle’s
quadrant.
Example
Find the reference angle and quadrant of the
following:
3
4

Or 45º
Reference Angle:
4
Quadrant of Angle: Second Quadrant
3  45  135
Stewart’s Table: Finding Exact Values of Trig
Functions
R.A.
Sin
Cos
0
0
0
2
1 1

2
2
2
2
1
3
2
1
2

6

4

3

2
3
2
2
2
4 2
 1
2
2
Each time the square root
number goes up by 1
0
Reverse the order of the
values from sine
Tan
1. Find the
value of the
Reference
Angle.
2. Find the
angles
quadrant to
figure out the
sign (+/-).
How to Remember which Trigonometric
Function is Positive
1
Just
Sine
S
-1
Just
Tangent
A
STUDENTS
ALL
TAKE
CALCULUS
T
C
-1
All
1
Just
Cosine
Example 1
Find the exact value of the following:
Thought process
cos  34 

Reference Angle:
4
Cosine of Reference Angle: cos  4  
2
2
Quadrant of Angle: 3  45  135  Second Quadrant
Sign of Cosine in Second Quadrant:
Therefore:
cos  34   
2
2
S
A
T
C
 Negative
The only thing required for a correct
answer (unless the question says explain)
Example 2
Find the exact solutions to the equation below if 0 ≤ x ≤ 2π:
Isolate the Trig
Function
2cos  x   1
cos  x    12
the answer in
x  cos   12  Finddegrees
first
x  2.094 or 120
1
Are there more
answers?
120°
1
Find the Reference
Angle
-1
60°
60°
Convert the answers to radians
1 Use the reference angle to
-1
x
120

 180
2
x
3
x
find where Cosine is also
negative
180°+60°
=240°
The answer must
be in Radians
2
3
,
4
3
240

 180
4
x
3
x
Example 3
Find the exact value of the following:
Thought process
tan  53 

Reference Angle:
3
Tangent of Reference Angle: tan  3   3
Quadrant of Angle: 5  60  300  Fourth Quadrant
Sign of Tangent in Fourth Quadrant:
 ,  
 negative
positive  Negative
 , 
Therefore:
tan  53    3
 ,  
 , 
The only thing required for a correct
answer (unless the question says explain)