 Relative acute angles

Lesson

In the unit circle centred at the origin of a coordinate system, we measure angles of any magnitude between the positive axis and the radius to a point that moves on the circle. The trigonometric functions of those angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant.

For example, by reference to a unit circle diagram, it can be checked that  $\sin135^\circ=\sin45^\circ$sin135°=sin45°. That is, the second quadrant angle $135^\circ$135° is related to the acute angle $45^\circ$45°. Similarly, $\cos225^\circ=-\cos135^\circ$cos225°=cos135° and we say $45^\circ$45° in the third quadrant is related to  $45^\circ$45° in the first quadrant.

The first quadrant or acute angle to which a particular angle is related in this way may be called a reference angle. The reference angle is between $0^\circ$0° and $90^\circ$90°.  (In some countries we also refer to the reference angle as the relative acute angle)

To find a reference angle, first, if necessary, add or subtract multiples of $360^\circ$360°  to obtain an angle between $0^\circ$0° and $360^\circ$360°. Then, decide what quadrant the angle is in.

If the resulting angle is in the first quadrant, it is the reference angle. If it is in the second quadrant, subtract it from $180^\circ$180° to obtain the reference angle. If it is in the third quadrant, subtract $180^\circ$180°  from the angle. In the fourth quadrant, subtract the angle from $360^\circ$360°

Example:

Find the acute reference angle for the angle $-534^\circ$534°

First, we add $2$2 lots of $360^\circ$360° to obtain $186^\circ$186°, an angle in the range $0^\circ$0° to $360^\circ$360°

The angle $186^\circ$186° is in the third quadrant. So, we subtract $180^\circ$180° from it. Hence, the reference angle is $6^\circ$6°.

More Examples

Example 2

Find the reference angle for $197^\circ$197°.

Example 3

Point $P$P on the unit circle shows a rotation of $330^\circ$330°.

What acute angle in the first quadrant can $330^\circ$330° be related to?

We want to evaluate $\tan\left(\left(-75\right)\right)$tan((75)) by first rewriting it in terms of the related acute angle $\theta$θ. What is the related acute angle of $\left(-75\right)^\circ$(75)°?