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Since y = cos -1x is the inverse of the function y = cos x, the function y = cos -1x if and only if cos y = x. But, since y = cos x is not one-to-one, its domain must be restricted in order that y = cos -1x is a function. To get the graph of y = cos -1x, start with a graph of y = cos x. |
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Restrict the domain of the function to a one-to-one region - typically |
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Reflect the graph across the line y = x to get the graph of y = cos-1 x (y = arccos x), the black curve at right. Notice that y = cos -1x has domain [-1, 1] and range . |
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| So, when you ask your calculator to graph y = cos -1x, you get the graph shown at right. (The viewing window is [-2, 2] x [-0.5, 3.5].) |  |
Evaluating cos -1x expressions follows the same procedure as evaluating sin -1x expressions - you must be aware of the domain and range of the function! Here is an example:
If y = cos -1(-1/2), then cos y = -1/2. This equation has an infinite number of solutions, but only one of them (
) is in the range of cos -1x. Thus:
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This is illustrated in the figure at right. The vertical red lines indicate some of the locations where y = -1/2, but only one (the solid red line) is within the domain of y = cos -1x (which is ![[0, pi]](Images/closed_int_zero2pi.gif){kind=small}
).
The derivative of cos-1x is: (The derivation is essentially the same as for sin -1x.)
A graph of y = cos-1x and its derivative is shown at right. Notice that since cos-1x is a strictly-decreasing function, its derivative is always negative.
Well, there aren't any! Since the derivatives of sin-1x and cos-1x are so similar (and the derivative of sin-1x is simpler), it is standard practice to say:
