Periodic functions > The sine function
123456The sine function

## Theory

Take a look at the applet: sine function

Above you see the graph of $f\left(x\right)=sin\left(x\right)$ with $x$ in radians on $\left[0,2\pi \right]$ . The solutions of $sin\left(x\right)=c$ are shown ( $c$ is a constant).

The solution of $sin\left(x\right)=c$ within $\left[-\frac{1}{2}\pi ,\frac{1}{2}\pi \right]$ is called the arcsine of $c$ : $x=arcsin\left(c\right)$ .
There (often) is another solution within a range of one period.
Due to the symmetry of the graph that other solution is $x=\pi -arcsin\left(c\right)$ .

Because of the period of $2\pi$ all solutions of $sin\left(x\right)=c$ are given by :
$x=arcsin\left(c\right)+k\cdot 2\pi \vee x=\pi -arcsin\left(c\right)+k\cdot 2\pi$ with $k$ any integer.

The equation $sin\left(x\right)=c$ only has solutions if $-1\le c\le 1$ .

There are some values that are convenient to use:

• $sin\left(0\right)=0$

• $sin\left(\frac{1}{6}\pi \right)=\frac{1}{2}$

• $sin\left(\frac{1}{4}\pi \right)=\frac{1}{2}\sqrt{2}$

• $sin\left(\frac{1}{3}\pi \right)=\frac{1}{2}\sqrt{3}$

• $sin\left(\frac{1}{2}\pi \right)=1$

and vice versa:

• $arcsin\left(0\right)=0$

• $arcsin\left(\frac{1}{2}\right)=\frac{1}{6}\pi$

• $arcsin\left(\frac{1}{2}\sqrt{2}\right)=\frac{1}{4}\pi$

• $arcsin\left(\frac{1}{2}\sqrt{3}\right)=\frac{1}{3}\pi$

• $arcsin\left(1\right)=\frac{1}{2}\pi$

You should use these values if exact values are required.