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.