Video: Finding Principal Angles

Given the angle (273πœ‹)/3, find the principal angle.

02:10

Video Transcript

Given the angle 273πœ‹ over three, find the principal angle.

First of all, what is the principal angle? If we have a coordinate plane and the initial arm of the angle is at zero degrees and then we have a terminal arm somewhere. The counterclockwise angle between the initial arm and the terminal arm is a principal angle if it is between zero and 360 degrees or between zero and two πœ‹ radians.

We’ve been given the angle that measures 273πœ‹ over three. The first thing we can do is go ahead and divide 273 by three. When we do that, we find out that this angle is 91πœ‹. We can tell by this measure that we’re going to be working in radians and not degrees. 91πœ‹ is larger than two πœ‹, which means it’s not a principal angle as it stands. We want to find out where would 91πœ‹ fall on a coordinate system. If we let its initial arm be at zero and then if the terminal arm was also at zero after one turn, it would be two πœ‹.

But we can also say that four πœ‹ would be located here, as would six πœ‹. At this point, we can think about 91πœ‹ in a different way. 91πœ‹ is the same thing as 90πœ‹ plus one πœ‹. And 90πœ‹ would be a terminal arm located along the π‘₯-axis. And if 90πœ‹ is located here, then 91πœ‹ would be located here. And if this is our angle πœƒ, to write 91πœ‹ as a principal angle would be πœ‹.

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