Finding the Exact Value of Trigonometric Functions

1
Finding the Exact Value of Trigonometric Functions
2
Review Special Right Triangles
Find the missing side Lengths
60
p / 3
1
45
p / 4
1
p / 6
30
p / 4
45
3
Important Points on Unit Circle
1
p / 2
90
p / 3
2p / 3
120
60
p / 4
3p / 4
135
45
5p / 6
p / 6
30
150
p
0
0
180
-1
1
11p / 6
7p / 6
330
210
7p / 4
5p / 4
315
225
5p / 3
4p / 3
240
300
3p / 2
270
-1
4
Important Points on Unit Circle
1
p / 2
p / 3
2p / 3
p / 4
3p / 4
5p / 6
p / 6
p
0
-1
1
11p / 6
7p / 6
7p / 4
5p / 4
5p / 3
4p / 3
3p / 2
-1
5
Important Points on Unit Circle
1
90
120
60
135
45
30
150
0
180
-1
1
330
210
315
225
240
300
270
-1
6
Important Points on Unit Circle
1
p / 2
p / 3
2p / 3
p / 4
3p / 4
5p / 6
p / 6
p
0
-1
1
11p / 6
7p / 6
7p / 4
5p / 4
5p / 3
4p / 3
3p / 2
-1
7
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.

Then multiply the number in the numerator by the
degree to find the angles quadrant.
8
Stewarts Table Finding Exact Values of Trig
Functions
R.A. Sin Cos Tan

1. Find the value of the Reference Angle.
2. Find the angles quadrant to figure out the sign
   (/-).

Each time the square root number goes up by 1
Reverse the order of the values from sine
9
Example 1

• Find the exact value of the following

10
Example 2

• Solve

p / 6
p / 6
11
Slope on the Unit Circle
What is the slope of the terminal side of an
angle on the unit circle?
1
(cos?,sin?)
sin?
Opposite
?
-1
1
cos?
Adjacent
Slope
-1
12
A Definition of Tangent
The tangent function is defined as

• There are values for which the tangent function
  are undefined

Any T that makes cos(T)0.
In general
For any integer n.
13
Stewarts Table Finding Exact Values of Trig
Functions
R.A. Sin Cos Tan

1. Find the value of the Reference Angle.
2. Find the angles quadrant to figure out /-.

Each time the square root number goes up by 1
Reverse the order of the values from sine
14
Example 3

• Find the exact value of the following

Thought process
The only thing required for a correct answer
(unless the question says explain)