Given the angle 273𝜋 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 radians.
We’ve been given the angle that measures 273𝜋 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. We can tell by this measure that we’re going to be working in radians and not degrees. 91𝜋 is larger than two 𝜋, which means it’s not a principal angle as it stands. We want to find out where would 91𝜋 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.
But we can also say that four 𝜋 would be located here, as would six 𝜋. At this point, we can think about 91𝜋 in a different way. 91𝜋 is the same thing as 90𝜋 plus one 𝜋. And 90𝜋 would be a terminal arm located along the 𝑥-axis. And if 90𝜋 is located here, then 91𝜋 would be located here. And if this is our angle 𝜃, to write 91𝜋 as a principal angle would be 𝜋.