### Video Transcript

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

We know that the principal angle is
the positive angle, so itβs 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, letβs ask ourselves, what does
negative 23π over five radians actually look like? Itβs negative, so itβs 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, weβre going to complete two
lots of full turns and another three-fifths π radians.

So, hereβs 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 weβre measuring in a
clockwise direction, itβs actually negative.

To find the angle which is
coterminal to this and positive, weβre 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.