Download Reference Angles

HA2: Notes Section 13.3:
TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES
Example 1: Find the exact values of the six trigonometric functions of  if the terminal side of
b) (3, -7).
 contains the point a) (8, 5)
Reference Angles
An important concept in Trigonometry is the idea of a reference angle. You will soon see its
importance, so for now let’s just look at what it is. The reference angle is the angle the
terminal side makes with the closest x-axis. It is always an acute angle, is always positive, is
called  and is between 0 and 90 degrees.
Examples:
Find the measure of the reference angle for each angle.
a. 150
b.  20
c.  545
d. 300
Rules for finding reference angles:
1. If your terminal side is in Quadrant I, then  is the same as your angle.
2. If your terminal side is in Quadrant II, then  is found by 180-(your angle)
3. If your terminal side is in Quadrant III, then  is found by (your angle) –180.
4. If your terminal side is in Quadrant IV, then  is found by 360-(your angle).
Complete the table to summarize the signs of the trigonometric functions for each quadrant
Quadrant
Quadrant
Quadrant
Quadrant
Function
I
II
III
IV
sin  or csc 
cos  or sec 
tan  or cot 
To find the trig values in all quadrants we use reference angles.
Steps:
1. To find values of angles in other quadrants, graph the angle and its terminal side.
2. Drop a vertical line with the closest x-axis to make a right triangle.
3. Identify your reference angle and that becomes your  value.
4. Label the sides of your triangle based on the ratios of the special triangles being careful
of negative values.
5. Find your trig value.
Examples: Find each value without using a calculator.
1. sin 45
2. cos 210
3. sin 300
4. tan 330

3
5
 11 
5. sin
6. sin
7. cos
8. tan  

4
4
6
 4 
7
29
 8 
9. tan  
10. sec
11. csc

4
3
 3 
How do you evaluate trig values on the x and y-axes? You can’t make a triangle, but you can
create a point. This will give you a value for x and y from which you can find r to evaluate your
trig functions.
Examples: Find each value without using a calculator.
1. sin 0
2. sin 180
3. cos180
4. tan  270

3
5. sin
6. cos
2
2
If the terminal side of angle  lies on one of the axes,  is called a quadrantal angle. What
are the quadrantal angles? List the values for each of the six trig functions at each
quadrantal angle.
Show the following: tan  
sin 
1
1
1
cos 

, csc  
, sec  
, cot  
cos 
sin 
cos 
tan  sin 
Examples:
1
and the terminal side is in quadrant 2(why do you need to know the
2
quadrant?). Find a. cos  , b. tan  , c. sec  .
1. Suppose sin  
2. Suppose cos  =3/5 and the terminal side is in quadrant 1. Find a. sin  , b. csc  , c. cot  .
3. Suppose sec  =2 and  is in quadrant 4. Find sin  , cos  , cot  .
4. Suppose  is an angle in standard position with the given conditions. State the quadrant or
quadrants in which the terminal side of  lie.
a. sin  >0
b. cos  >0
c. sin  >0 and cos  <0
5. Find two values of x that make each statement true.
a. cos  =1
b. cos  =0 c. sin  =undefined d. tan  =0
e. tan  =undefined
6. Tell whether the value of each trigonometric function is positive, negative, zero, or undefined.

11
a. sin
b. tan 315 c. sin
d. cos 450
2
4