Given the angle 39𝜋 over four, find the principal angle.
We will begin by sketching the unit circle and identifying where the angle 39𝜋 over four lies. We recall that positive angles in standard position are measured in a counterclockwise direction from the positive 𝑥-axis. There are two 𝜋 radians in a complete circle. So we can mark on 𝜋 over two, 𝜋, three 𝜋 over two, and two 𝜋 radians as shown. 39𝜋 over four is equal to nine and three-quarter 𝜋. This can also be written as eight 𝜋 plus seven 𝜋 over four.
We know that eight 𝜋 is equal to four complete turns. Seven 𝜋 over four lies between three 𝜋 over two and two 𝜋. This means that our angle lies in the fourth quadrant as shown. We are asked to find the principal angle. We know that if 𝜃 is an angle in standard position as in this case, then the counterclockwise angle between the initial and terminal side of 𝜃, which must be less than a full turn, is called the principal angle of 𝜃. This means that the principal angle is equal to seven 𝜋 over four.