Simplify sin of 360 degrees minus 𝜃.
To do this, let’s consider the unit circle. We’ll let 𝜃 be an angle in standard position. And the point 𝑥, 𝑦 will be the coordinates of the point of intersection between the unit circle and the terminal side of 𝜃. The sin of 𝜃 will equal 𝑦 over one. Therefore, we can label this distance as sin of 𝜃. Now we want to know where 360 degrees minus 𝜃 would fall on our unit circle. The angle 360 degrees minus 𝜃 in a counterclockwise direction would look like this. To calculate the sin of 360 degrees minus 𝜃, we would use its reference angle.
Here, we have a reflection in the 𝑥-axis. Recall that we’ve already said 𝑦 equals the sin of 𝜃. Therefore, negative 𝑦 equals the negative sin of 𝜃, which is also equal to the sin of 360 degrees minus 𝜃. This illustrates a principle that we know about the related angle identities for trigonometric functions. And that is that the sin of 360 degrees minus 𝜃 is equal to the negative sin of 𝜃. We’ve been able to simplify sin of 360 degrees minus 𝜃 and show that it’s equal to negative sin of 𝜃.