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.