Simplify the cos of 360 degrees minus 𝜃.
In order to simplify the trigonometric expression in this question, we will consider related angles in the unit circle. We recall that positive angles are measured in the counterclockwise direction from the positive 𝑥-axis as shown. If we let point 𝑃 lie on the unit circle such that 𝑂𝑃 makes an angle of 𝜃 degrees with the positive 𝑥-axis, we know that 𝑃 has coordinates cos 𝜃, sin 𝜃. Next, let’s consider the point 𝑄 that also lies on the unit circle, where the angle 𝑂𝑄 makes with the positive 𝑥-axis in the counterclockwise direction is 360 degrees minus 𝜃. This means that 𝑄 has coordinates cos 360 degrees minus 𝜃, sin 360 degrees minus 𝜃.
By the symmetry of the unit circle, the 𝑥-coordinates of points 𝑃 and 𝑄 are equal, and the cos of 360 degrees minus 𝜃 must therefore be equal to the cos of 𝜃. Whilst it is not required in this question, we also see that the sin of 360 degrees minus 𝜃 is equal to negative sin 𝜃, since the 𝑦-coordinates of points 𝑃 and 𝑄 are the negative of one another. The answer to this question, however, is that the cos of 360 degrees minus 𝜃 is equal to cos 𝜃. This is one of our related angle identities that we need to recall.