Question Video: Finding Principal Angles Mathematics • First Year of Secondary School

Video Transcript

Given the angle negative two 𝜋 by three, find the principal angle.

We’re asked to determine the principal angle of negative two 𝜋 by three. We start by recalling that the principal angle will be an angle in standard position with the same terminal side and our angle should be between zero and two 𝜋. Therefore, to find the principal angle, we’ll start by sketching negative two 𝜋 by three in standard position. First, since our angle is negative, this means we’re going to need to measure our angle clockwise from the positive 𝑥-axis. Next, it might help us to measure each of the right angles. Going clockwise from the positive 𝑥-axis, that’s negative 𝜋 by two, negative 𝜋, and negative three 𝜋 by two.

Next, either by using the coefficients or calculating, we can see that negative two 𝜋 by three lies between negative 𝜋 by two and negative 𝜋. In other words, its terminal side lies in the third quadrant. We want to find the principal angle. Remember, this is measured in standard position and it’s between zero and two 𝜋. So it’s measured clockwise. And it needs to have the same terminal side. This gives us the following sketch. And together with the measure of negative two 𝜋 by three, we can see this makes a full revolution. In other words, the principal angle plus two 𝜋 by three must be equal to two 𝜋.

And we can then solve for the principal angle by subtracting two 𝜋 by three from both sides of the equation. We get that the principal angle will be two 𝜋 minus two 𝜋 by three, which we can evaluate by noting that two 𝜋 is equal to six 𝜋 by three. We subtract two 𝜋 by three from this to get four 𝜋 by three, which is our final answer. Therefore, we were able to show the principal angle of negative two 𝜋 by three is four 𝜋 by three.