Question Video: Finding Principal Angles Mathematics

Given the angle βˆ’2πœ‹/3, find the principal angle.

01:40

Video Transcript

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

We’re asked to determine the principal angle of negative two πœ‹ 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œ‹. Therefore, to find the principal angle, we’ll start by sketching negative two πœ‹ 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 π‘₯-axis. Next, it might help us to measure each of the right angles. Going clockwise from the positive π‘₯-axis, that’s negative πœ‹ by two, negative πœ‹, and negative three πœ‹ by two.

Next, either by using the coefficients or calculating, we can see that negative two πœ‹ by three lies between negative πœ‹ by two and negative πœ‹. 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œ‹. 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œ‹ by three, we can see this makes a full revolution. In other words, the principal angle plus two πœ‹ by three must be equal to two πœ‹.

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

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