FINDING THE MEASURE OF THE REFERENCE ANGLE

Example : 

What is the reference angle for each angle in standard position? 

(a)   170°      (b)   345°     (c)   72°   (d)   215°

Solution :

(a) 170°

The reference angle of 170 is 190 - 170. That is 20°.

(b) 345°

The reference angle of 345 is 360 - 345. That is 15°.

(c)   72°   

The reference angle of 72 is 90 - 72. That is 18°.

(d)   215°

The reference angle of 215 is 270 - 215. That is 55°.

Finding Other Angles in Other Quadrant with Given Reference Angles

To determine the other angles in other quadrants, we have to use the following table.

The reference angle always be lesser than 90 degree.

Example :

Determine the measure of the three other angles in standard position, 0° < θ < 360°, that have a reference angle of

(a) 45°       (b) 60°       (c) 30°       (d) 75°

Solution :

Let θ be given angle.

Required angle in 2^nd quadrant  =  180 - θ 

=  180 - 45  =  135

Required angle in 3^rd quadrant  =  180 + θ 

  =  180 + 45  =  225

Required angle in 4^th quadrant  =  360 - θ 

  =  360 - 45  =  315

Hence the required angles are 135, 225 and 315.

(b) 60°  

Solution :

Let θ be given angle.

Required angle in 2^nd quadrant  =  180 - θ 

=  180 - 60  =  120

Required angle in 3^rd quadrant  =  180 + θ 

  =  180 + 60  =  240

Required angle in 4^th quadrant  =  360 - θ 

  =  360 - 60  =  300

Hence the required angles are 120, 240 and 300.

(c) 30°  

Solution :

Let θ be given angle.

Required angle in 2^nd quadrant  =  180 - θ 

=  180 - 30  =  150

Required angle in 3^rd quadrant  =  180 + θ 

  =  180 + 30  =  210

Required angle in 4^th quadrant  =  360 - θ 

  =  360 - 30  =  330

Hence the required angles are 150, 210 and 330.

(d) 75°  

Solution :

Let θ be given angle.

Required angle in 2^nd quadrant  =  180 - θ 

=  180 - 75  =  105

Required angle in 3^rd quadrant  =  180 + θ 

  =  180 + 75  =  255

Required angle in 4^th quadrant  =  360 - θ 

  =  360 - 75  =  285

Hence the required angles are 105, 255 and 285.

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