In trigonometry, when we discuss the rotation of points on the unit circle, we often talk about angles in terms of radians. Since a complete rotation around the unit circle is \(2\pi\) radians, any rotation can be extended by adding or subtracting multiples of \(2\pi\) radians to yield an equivalent position on the unit circle.
Given that point A is at 0 radians with coordinates (1,0) and that point B is the result of point A rotating by \(r\) radians, there are infinitely many angles of rotation that could take A to the same position as B. This is because you can rotate around the circle as many times as you like and still end up at point B.
To name two specific other angles of rotation that take A to B:
1. One angle could be the initial rotation \(r\) plus one full rotation around the circle, which is \(r + 2\pi\) radians. This is a positive angle that effectively rotates the point to B, then goes around the entire circle once and ends up back at B.
2. For the negative angle, we could go in the reverse direction (clockwise). So, we subtract one full rotation from \(r\). This gives us \(r - 2\pi\) radians. This rotation takes the point to B and then continues in the reverse direction all the way around the circle back to B.
So, the two other angles of rotation that take A to B are \(r + 2\pi\) and \(r - 2\pi\). The mathematical reasoning behind this is based on the periodic nature of the trigonometric functions and the fact that rotating a point by \(2\pi\) radians (or full multiples thereof) does not change its position on the unit circle.
