The reference angle is the acute angle formed between the terminal arm and the x-axis. The reference angle is always positive and measures between 0° and 90°.
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°.
To determine the other angles in other quadrants, we have to use the following table.
The reference angle always be lesser than 90 degree.
Angles in quadrants 2nd quadrant 3rd quadrant 4th quadrant |
Formula 180 - given angle 270 - given angle 360 - given angle |
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 2nd quadrant = 180 - θ
= 180 - 45 = 135
Required angle in 3rd quadrant = 180 + θ
= 180 + 45 = 225
Required angle in 4th 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 2nd quadrant = 180 - θ
= 180 - 60 = 120
Required angle in 3rd quadrant = 180 + θ
= 180 + 60 = 240
Required angle in 4th 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 2nd quadrant = 180 - θ
= 180 - 30 = 150
Required angle in 3rd quadrant = 180 + θ
= 180 + 30 = 210
Required angle in 4th 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 2nd quadrant = 180 - θ
= 180 - 75 = 105
Required angle in 3rd quadrant = 180 + θ
= 180 + 75 = 255
Required angle in 4th quadrant = 360 - θ
= 360 - 75 = 285
Hence the required angles are 105, 255 and 285.
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