In this discussion, we learn how to graph inverse sine and identify its domain and range. Here is an outline of this discussion:
- Begin with the Graph of the Sine Function
- Restrict the Domain of Sine
- On a New Graph, Do the Following:
- Draw a Neat Number Quadrant
- Draw the Line y = x
- Draw the Restricted Graph of Sine
- Swap the x and y Values
- Reflect the New Graph about the Line y = x
- Observe the Domain and Range of Inverse Sine
Let us begin!
Graph the Sine Function
First, observe the graph of sine.
Starting from zero, and traveling to the right, we know that:
sin(0) = 0
\sin(\frac{\pi}{2}) = 1
\sin(\pi) = 0
\sin(\frac{3\pi}{2}) = -1
\sin(2\pi) = 0
Starting from zero, and traveling to the left, we know that:
sin(0) = 0
\sin(-\frac{\pi}{2}) = -1
\sin(-\pi) = 0
\sin(-\frac{3\pi}{2}) = 1
\sin(-2\pi) = 0
Draw the graph of sine, connecting the dots that are mentioned above.
We know that sine is a function because it passes the vertical line test. This means that, if any vertical line is drawn to the graph, it will only intersect the graph one time.
However for a function to have an inverse, it must be one-to-one. In other words, the graph must pass the horizontal line test. This means that if any horizontal line is drawn through the graph, it must touch the graph one time only.
Clearly, the sine function fails the horizontal line test!
Restrict the Domain of Sine
Mathematicians are clever! Mathematicians restricted the domain from -\frac{\pi}{2} to \frac{\pi}{2}. In this way, the graph passes the Horizontal-Line-Test.
Thus, sine has an inverse. Take a closer look at the restricted graph of sine.
Steps to Graphing Inverse Sine
Step 1: Draw a Neat Number Quadrant
To draw the graph of inverse sine, always begin with a neat number quadrant. When placing x and y values on the graph, it is important to note that \frac{\pi}{2} is approximately 1.6. Thus, put \frac{\pi}{2} about mid-way between the numbers 1 and 2.
Step 2: Draw the Line y = x
Second, draw the line y = x. I use the points (0,0), (1,1), and (-1,-1) and then draw a dotted line through the points.
Step 3: Draw the Restricted Graph of Sine
Step 4: Swap the x and y Values
Next, swap the x and y values. Put dots where the new points will be located.
When swapping the x and y values, note: (\frac{\pi}{2},1)\;\longrightarrow\;(1,\frac{\pi}{2}),
(0,0)\, becomes\, (0,0),
(-\frac{\pi}{2},-1)\;\longrightarrow\;(-1,-\frac{\pi}{2}).
Step 5: Draw the New Graph by Reflecting about the line y = x.
Now we connect the dots and reflect about the line y = x to get the new graph.
Separating the new graph from the old one, the graph of inverse sine is obtained.
Observe the Domain and Range of Inverse Sine
Now the domain and range of inverse sine can be identified. The domain of inverse sine is
[-1, 1]
This means that, given a function in the form y = \sin^{-1}(x) ,
the x-value must fall within the interval [-1,1]. The Range of inverse sine is
\left[-\frac{\pi}{2},\frac{\pi}{2}\right]
This means that, given a function in the form
y = \sin^{-1}(x) ,
the y-value must fall within the interval
\left[-\frac{\pi}{2},\frac{\pi}{2}\right].
On the unit circle, only choose values from the right region when considering the range of inverse sine.
