# How can you describe the values of $x$ for which $\sin(x)$ has the same output?

I want to describe all the values of $x$ for which $\sin(x)$ has the same output.

Example: at $x=0,\, x=\pi,\, x=2\pi$ we have $\sin(x)=0$.

For other values of $x$ the similarity seems to be more complicated.

I tried to figure it out, and came up with $\pi - x +2k\pi$. Does that make sense?

• $x+2\pi k, \pi-x+2\pi k$ – David Peterson Nov 3 '16 at 7:54
• it is odd and $2\pi$ periodic – reuns Nov 3 '16 at 8:00
• $\dfrac\pi2\pm\left(\dfrac\pi2-x\right)+2k\pi$. – Yves Daoust Nov 3 '16 at 8:22

Draw a trigonometric cirle in the plane. You know that the sine of an angle $\alpha$ is the $y$-coordinate of the point $P$ in the plane obtained intersecting one side of your angle with the circle. Thus, the angles having the same sine (up to an additive constant of $2k\pi$, $k\in\mathbb{Z}$, because of periodicity) are precisely those you obtain intersecting a horizontal line with your circle. Now, there are three possible cases:

1) no intersection: your line is of the kind $y=c$, with $c>1$ or $c<-1$.

2) one intersection: consider $y=1$ and $y=-1$. These give you the (countably infinite) angles $\alpha=\pm {\pi\over 2}+2k\pi$.

3) two intersections: for $y=c$ with $-1<c<1$ you get the angles $\alpha+2k\pi$ and $\pi-\alpha+2k\pi$, which have the same sine.

There are two trains (series of regularly spaced/periodic numbers)

$$x_1, 2 \pi + x_1,4 \pi + x_1, \quad \dots \quad \pi - x_1,3 \pi - x_1, 5 \pi - x_1 \dots$$

A single general formula clubs them together.

$$x_1 +(-1)^n 2 n \pi.$$

If you draw a graph of $y=\sin x$ and a line parallel to x-axis.. $y_1 = \sin x_1,$ you can see the two trains separately at all cutting points.