If $x=r\cos(\theta)$ then in $?=r\cos(\theta+a)$ what is $?$ equal to?

Question

What I mean by $r\cos(\theta+a)$ is that it's the same function $r\cos(\theta)$ but it's translated by $a$ units, if this makes any sense. I just want to know what it means in terms of $x$.

Answer 1

This question has a simple and slightly confusing answer:

The answer is simply that you took a point $P(x,y)$ and rotated it $a$ radians (or degrees) around the origin.

Another answer I am inclined to give you is the following:

$$r\cos(\theta+a)=r[\cos(\theta)\cos(a)-\sin(\theta)\sin(a)]$$

$$=r\cos(\theta)\cos(a)-r\sin(\theta)\sin(a)$$

$$=x\cos(a)-y\sin(a)$$

If you recall, $r\sin(\theta)=y$, and, as you will notice, $\cos(a)$ and $\sin(a)$ are both constants.

Answer 2

Hint. $$r\cos(\theta + a) = r\left[\cos(\theta)\cos(a)-\sin(\theta)\sin(a) \right]$$