Problem 61 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$-765^{\circ}$$

Step-by-Step Solution

Answer

The positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with \(-765^{\circ}\) is \(315^\circ\).

Step 1: Calculation of Equivalent Positive Angle

First convert the negative angle to an equivalent positive angle. To do that, continuously add \(360^\circ\) until the resulting angle is positive and less than \(360^\circ\). It helps to note that \(-765^\circ + 360^\circ = -405^\circ\) and \(-405^\circ + 360^\circ = -45^\circ\). So, initially, \( -765^\circ \) is equivalent to \( -45^\circ \) as they share the same terminal side.

Step 2: Conversion to Positive Coterminal Angle

Since the angle must be positive and less than \(360^\circ\), converting the \(-45^\circ\) to a positive angle that fulfills this condition will be done by adding \(360^\circ\). So, \(-45^\circ + 360^\circ = 315^\circ\). Therefore, \( 315^\circ \) is the positive angle that is less than \(360^\circ\) and is also coterminal with \(-765^\circ\).