1. 1.4 Reference Angles

2. Definition:
Reference Angle -
A positive acute angle found between the terminal side of an angle in standard position and the x-axis.

3. Examples:
Determine the reference angle of the following angles:
 = 65
Let’s draw a picture:
Let’s calculate the reference angle…
Terminal side
Reference Angle
’ = 65
Let’s see where the reference angle is…

4. Examples:
Determine the reference angle of the following angles:
 = 192
Let’s draw a picture:
Let’s calculate the reference angle…
Reference Angle
’ = 192- 180
’ = 12
Terminal side
Let’s see where the reference angle is…

5. Examples:
Determine the reference angle of the following angles:
 = 119
Let’s draw a picture:
Let’s calculate the reference angle…
Terminal side
’ = 180- 119
Reference Angle
’ = 61
Let’s see where the reference angle is…

6. Examples:
Determine the reference angle of the following angles:
 = 341
Let’s draw a picture:
Let’s calculate the reference angle…
Reference Angle
’ = 360- 341
Terminal side
’ = 19
Let’s see where the reference angle is…

7. Examples:
Determine the reference angle of the following angles:
 = /9
Let’s draw a picture:
Let’s calculate the reference angle…
Reference Angle
Terminal side
’ = /9
Let’s see where the reference angle is…

8. Examples:
Determine the reference angle of the following angles:
 = 7/5
Let’s draw a picture:
Let’s calculate the reference angle…
Reference Angle
’ = 7/5 - 
’ = 2/5
Terminal side
Let’s see where the reference angle is…

9. Examples:
Determine the reference angle of the following angles:
We need to first find a coterminal angle:
 = -3/8
Let’s draw a picture:
-3/8 + 2
-3/8 + 16/8
13/8
Let’s calculate the reference angle…
Reference Angle
Terminal side
Let’s see where the reference angle is…
’ = 2 - 13/8
’ = 3/8

10. 1.4 Working with Reference Angles

11. Now that you have been exposed to the concept of reference angle, it is time to finish this section.
We can now determine trigonometric values of non-acute angles.

12. Example:
Let (-3,4) be a point on the terminal side of  (in standard position). Determine the sine, cosine and tangent of .
Step 1: Draw a picture of the situation.
Step2: Draw the reference triangle (drop the altitude to the x-axis). Then determine the lengths of the sides of the reference triangle.
Step 3: Use the non-unit circle definitions to determine the values of the trig. functions requested.
Determine the radius of the circle passing through the point referenced.
(-3,4)  5 4 3
sin = cos = tan = 4 5 -3 5 -4 3

13. Example:
Let (-4,-6) be a point on the terminal side of  (in standard position). Determine the sine, cosine, tangent, cosecant, secant and cotangent of .
Step 1: Draw a picture of the situation.
Step2: Draw the reference triangle (drop the altitude to the x-axis). Then determine the lengths of the sides of the reference triangle.
Step 3: Use the non-unit circle definitions to determine the values of the trig. functions requested.
Determine the radius of the circle passing through the point referenced. 4
sin = cos = tan =
csc = sec = cot =
-3√13 13
-2√13 13 3 2 6
(-4,-6) 
2√13
-√13 3 2 3
-√13 2

14. All Star Trig. Class
(−,+)
(+,+)
(−,−)
(+,−)