Chapter 4.3, Problem 2E

Solution

To identify: the one angle that is not co terminal with all the others.

  $-\frac{5\pi }{3}$

Given information:

The angles:

  $\frac{5\pi }{3},-\frac{5\pi }{3},\frac{11\pi }{3},-\frac{7\pi }{3},\frac{365\pi }{3}$

Definition Used:

Co-terminal angles:

Two angles in the expanded angle-measurement system, which has the same initial side and the same terminal side, yet have different measures are called co-terminal angles.

Angles are co-terminal whenever they differ by an integer multiple of $360°$ or by an integer multiple of $2\pi$ rad.

Explanation:

To find a co-terminal angle for the given angle:

If the angle is given in degrees, then add or subtract the multiples of $360°$ .

If the angle is given in radians, then add or subtract the multiples of  $2\pi$ .

Now, let $\frac{5\pi }{3}$ be the angle and try to find the possible co-terminal angles.

  $\begin{array}{c}-\frac{5\pi }{3}+2\pi =-\frac{5\pi }{3}+\frac{6\pi }{3}\\ =\frac{\pi }{3}\end{array}$

  $\begin{array}{c}\frac{11\pi }{3}-2\pi =\frac{11\pi }{3}-\frac{6\pi }{3}\\ =\frac{5\pi }{3}\end{array}$

  $\begin{array}{c}-\frac{7\pi }{3}+4\pi =-\frac{7\pi }{3}+\frac{12\pi }{3}\\ =\frac{5\pi }{3}\end{array}$

  $\begin{array}{c}\frac{365\pi }{3}-120\pi =\frac{365\pi }{3}-\frac{360\pi }{3}\\ =\frac{5\pi }{3}\end{array}$

So, it is clear that all the angles other than the angle $-\frac{5\pi }{3}$ are co-terminal.