Review question

# Which points in the $x$-$y$ plane satisfy $\sin y=\sin x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7374

## Solution

The graph of all those points $(x, y)$ in the $xy$-plane which satisfy the equation $\sin(y) = \sin(x)$ is drawn in

Firstly, we know that the equation is trivially true when $x=y$, and so we can rule out graphs (b) and (d) as the line $y=x$ is missing.

Secondly, we know $y = \pi - x$ is a solution to the equation, and this line does not appear in (d), which means that by elimination, (c) is the answer.

A fuller answer might run as follows: we know that for all values of $x$, $\sin(x)=\sin(\pi-x)$ and $\sin(x)=\sin(x + 2n\pi),$ where $n$ is an integer.

Thus if $\sin(y) = \sin(x)$ then $y = x + 2n\pi,$ OR $y = \pi - x + 2n\pi = -x +(2n-1)\pi.$

So the solution is given by the lines $\cdots, y=x-2\pi, y = x, y = x+2\pi, y = x + 4\pi, \cdots,$ together with the lines $\cdots, y = -x-\pi, y=-x+\pi, y=-x+3\pi, \cdots.$

These are the lines shown in (c).