Question Video: Finding Principal Angles | Nagwa Question Video: Finding Principal Angles | Nagwa

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

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Given the angle β2π/3, find the principal angle.

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Video Transcript

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

Weβ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, weβll start by sketching negative two π by three in standard position. First, since our angle is negative, this means weβ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, thatβ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 itβs between zero and two π. So itβ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.

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