In this discussion, we will learn how to graph inverse cosine and identify the domain and range.
Begin with the parent graph of cosine.
Clearly, cosine is a function because it passes the vertical line test.
This means that, if any vertical line is drawn to the graph in its upright position,
it will intersect the graph only once.
However, for a function to have an inverse, it must be one-to-one.
In other words, it must pass the horizontal line test.
This means that, if any horizontal line is drawn to the graph in its upright position,
it must touch the graph only once.
Clearly, the cosine function fails the horizontal line test!
Restrict\ the\ Domain\ from \ 0\ to\ \pi
However, mathematicians are clever!
Mathematicians \ Restricted\ the\\ Domain\ from \ 0\ to\ \pi
In this way, the graph passes the horizontal line test!
Thus, cosine has an inverse.
Take\ a\ closer\ look\ at\ the\ restricted\\ graph\ of\ cosine\ on\ the \ interval\\ from \ 0\ to\ \pi
Steps to Graphing Inverse Cosine
Step 1: Draw a Neat Number Quadrant
When placing your x and y values on the graph,
it is important to note that \frac{\pi}{2} is approximately 1.6
and \pi is approximately 3.14.
Thus, when drawing a cartesian plane, put \frac{\pi}{2} about mid-way between 1 and 2
and \pi at about the number 3.
Step 2: Draw the Line y = x
Draw a dotted line through the points (0,0), (1,1), and (-1,-1).
This is the line y = x.
Step 3: Draw the Restricted Graph of Cosine
Include the restricted graph of cosine on the graph.
Step 4:
Swap the x and y-values
When swapping the x and y values, the points:
(\frac{\pi}{2}, 0) becomes (0, \frac{\pi}{2}),
(0,1) becomes (1,0),
(\pi, -1) becomes (-1, \pi)
Step 5:
Draw the New Graph by Reflecting about the Line y = x
Last, connect the dots and reflect about the line y = x.
Separating the new graph from the old graph,
the graph of inverse cosine is obtained.
Observe the Domain and Range of Inverse Cosine
Now the domain and range of inverse cosine can be identified.
The domain of inverse cosine is [-1,1].
This means that, given a function in the form
y = cos-1(x),
the x-value must fall within the interval [-1,1].
The \ Range \ of \ inverse \ cosine \ \\is\ [0,\pi].
This means that, if you have a function in the form
y = cos-1(x),
the y-value must fall within the interval [0,\pi].
On the unit circle, only choose values from the upper region when considering the range of inverse cosine.
Now you are prepared to evaluate inverse cosine, which is the next discussion!
Are you ready? Go to the next article!
Inverse Cosine | How to Find the Exact Value
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practice problems with the answers, quizzes with the answers,
and more regarding inverse trig functions, consider taking my Math Course on Inverse trig functions!
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Angela Morgan