Question Video: Finding Principal Angles Mathematics

Given the angle βˆ’23πœ‹/5, find the principal angle.

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Video Transcript

Given the angle negative 23πœ‹ over five, find the principal angle.

We know that the principal angle is the positive angle, so it’s measured in a counterclockwise direction and it has a value in the closed interval from zero to two πœ‹ radians. And so, our job is to find the coterminal angle to negative 23πœ‹ over five which has a positive measure and lies in this interval. So, let’s ask ourselves, what does negative 23πœ‹ over five radians actually look like? It’s negative, so it’s going to be measured in a clockwise direction. And 23 over five is equivalent to four and three-fifths. And we know that a full turn is two πœ‹ radians. So, we’re going to complete two lots of full turns and another three-fifths πœ‹ radians.

So, here’s one full turn for two πœ‹ radians. Then, we complete a second full turn, and that takes us to four πœ‹ radians. And then, we have three-fifths, which is a little bit over one-half. And so, an angle that measures three-fifths πœ‹ radians will look a little something like this. Now, of course, three-fifths πœ‹ is between zero and two πœ‹. But because we’re measuring in a clockwise direction, it’s actually negative.

To find the angle which is coterminal to this and positive, we’re going to measure from the initial side to the terminal side in a counterclockwise direction like this. And so, the size of this angle is found by subtracting three πœ‹ over five from two πœ‹. By writing these numbers with the same denominator, we could write this as 10πœ‹ over five and then subtract their numerators to get seven πœ‹ over five. And so, given an angle of negative 23πœ‹ over five, the principal angle is seven πœ‹ over five.

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