Finding all the angles between \(0^\circ\) and \(360^\circ\)

Using the fact that each ratio is positive (or negative) in two quadrants, if you are asked to find the angle you expect to have two answers.

For example: Find \(\theta\) between \(0^{\circ}\leq \theta < 360^\circ\) such that \(\sin\theta=0.3\).

This example has two solutions as \(\sin\) is positive in two quadrants, the first quadrant and the second quadrant.

Calculator gives: \[ \theta=\sin^{-1} 0.3\approx 17.46^{\circ} \]

To get the angle in the second quadrant: \[ \theta =180^{\circ}-17.46^{\circ}=162.54^{\circ} \]

The two solutions can be seen in the below figure: