Reference Angle

Reference angle: 

Let  θ be an angle in standard position. Its reference angle θ’ is the acute angle  formed by the terminal side of  θ and the horizontal axis.

 

The values of the trigonometric functions of angles greater than (or less than ) can be determined from their values at corresponding acute angles called reference angles.

 

Below are the rules to find reference angles of θ in quadrants II , III and IV .

 

Quadrants	Formula   Radians                                   Degrees
II	π-θ                                        180-θ
III	θ-π                                       θ-180
IV	2π-θ                                       360-π

 

 

 

 

Lets look at this video lesson :

 

 

Example: Find reference angle θ’ of  each of the following angles.

      a)315°                       b) -150°                        c) 1.8°

Solution:  a) 315°

Since it lie in 4^th  quadrant  so we  use  rule  θ’=360-θ

Reference angle  is θ’=360-315 = 45°

b) -150°

First we find positive  coterminal angle of  -150

Positive coterminal angle = -150+360= 210

Since 210 lie in 3rd  quadrant  so we  use  rule θ’ = θ – 180

Reference angle  is θ’ = 210-180 =30°

c) 1.8

This angle is given in radians and it lie in 2nd quadrant .

Considering  π= 3.14

π/2   = 3.14/2  =1.57

1.8 lie between 1.57 and 3.14  which is 2^nd quadrant (π/2<θ<π )

For 2^nd quadrant  we use θ’ = π-θ

= 3.14-1.8 = 1.34 radians

 

 

 

Practice problems:   

Find reference angle of  each of the following angles.

• 57°
• -210°
• 1.2
• 11π/6
• 3.5

 

 

 

 

 

 

 

Answers:

• 57°
• 30°
• 1.2 radians
• π/6
• 0.36 radians