The Inverse of the Sine Function (Arcsin)

The inverse of the basic sine function is the arcsine function. The arcsine function relates the measure of the angles (in radians) of the unit circle with the |y|-values of the circle’s points.

The rule of the basic arcsine function is |f(x)=\arcsin (x).| The function can also be written |f(x)=\sin^{-1}(x).|

Note: Do not confuse the notation |\sin^{-1}(x)| with |\dfrac{1}{\sin (x)}.|

The inverse of a sine function is not a function. To become one, its range must be restricted.

The basic sine function is shown on the following Cartesian plane. To graph the inverse, exchange the |x|- and |y|-coordinates of the function’s points. A reflection of the points can also be conducted relative to the line |y=x.| For example, point |\left(\dfrac{\pi}{2},1\right)| becomes point |\left(1,\dfrac{\pi}{2}\right)\!.|

Another curve is then obtained that is not a function. Restricting the range of the inverse to the interval |\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]\!,| produces the arcsine function.

1. Interchange |x| and |y| in the rule.

2. Isolate the |y| variable.

3. Give the inverse rule.

Determine the rule for the inverse of the function |f(x)=-2\sin\left(\dfrac{1}{3}(x-\pi)\right)-5.|

1. Interchange |x| and |y| in the rule
   ||\begin{align}\color{#3B87CD}y&=-2\sin\left(\dfrac{1}{3}(\color{#FF55C3}x-\pi)\right)-5\\ \color{#FF55C3}x&=-2\sin\left(\dfrac{1}{3}(\color{#3B87CD}y-\pi)\right)-5\end{align}||

2. Isolate the |y| variable
   ||\begin{align}x&=-2\sin\left(\dfrac{1}{3}(\color{#3B87CD}y-\pi)\right)-5\\x+5&=-2\sin\left(\dfrac{1}{3}(\color{#3B87CD}y-\pi)\right)\\ \dfrac{x+5}{-2}&=\sin\left(\dfrac{1}{3}(\color{#3B87CD}y-\pi)\right)\end{align}||
   To isolate |y,| eliminate |\sin| by performing the inverse operation, |\sin^{-1}.|
   ||\begin{align}\color{#EC0000}{\sin^{-1}\!\left(\color{black}{\dfrac{x+5}{-2}}\right)}&=\dfrac{1}{3}(\color{#3B87CD}y-\pi)\\3\sin^{-1}\!\left(\dfrac{x+5}{-2}\right)&=\color{#3B87CD}y-\pi\\3\sin^{-1}\!\left(\dfrac{x+5}{-2}\right)+\pi&=\color{#3B87CD}y\end{align}||
   It is possible to simplify by working within the brackets. In fact, dividing by |-2| can also be written as multiplying by |-\dfrac{1}{2}.|||\begin{align}3\sin^{-1}\!\left(\color{#EC0000}{\dfrac{x+5}{-2}}\right)+\pi&=y\\ 3\sin^{-1}\!\left(\color{#EC0000}{-\dfrac{1}{2}(x+5)}\right)+\pi&=y\end{align}||

3. Give the inverse rule
   The inverse rule of the sine function is the following.
   ||f^{-1}(x)=3\sin^{-1}\!\left(-\dfrac{1}{2}(x+5)\right)+\pi||

Note: For the inverse to become a function, its range must be restricted.