# Question Video: Finding Principal Angles Mathematics

Given the angle −23𝜋/5, find the principal angle.

01:58

### Video Transcript

Given the angle negative 23𝜋 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 radians. And so, our job is to find the coterminal angle to negative 23𝜋 over five which has a positive measure and lies in this interval. So, let’s ask ourselves, what does negative 23𝜋 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 radians. So, we’re going to complete two lots of full turns and another three-fifths 𝜋 radians.

So, here’s one full turn for two 𝜋 radians. Then, we complete a second full turn, and that takes us to four 𝜋 radians. And then, we have three-fifths, which is a little bit over one-half. And so, an angle that measures three-fifths 𝜋 radians will look a little something like this. Now, of course, three-fifths 𝜋 is between zero and two 𝜋. 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 over five from two 𝜋. By writing these numbers with the same denominator, we could write this as 10𝜋 over five and then subtract their numerators to get seven 𝜋 over five. And so, given an angle of negative 23𝜋 over five, the principal angle is seven 𝜋 over five.

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