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Chapter 1: Problem 12

Sketch the graphs of \(y=\cos x\) and \(y=\cos ^{-1} x\) on the same set of axes.

Short Answer

Expert verified
Answer: The graph of \(y=\cos x\) is a periodic function with a period of \(2\pi\), with local maxima, local minima, and zeros at key points. The graph of \(y=\cos^{-1} x\) is defined for values of \(x\) between -1 and 1 and has outputs in the interval \([0, \pi]\). The graph of \(y=\cos^{-1} x\) is a reflection of the graph of \(y=\cos x\) over the line \(y = x\).

Step by step solution

01

Sketch \(y=\cos x\)

To sketch the graph of \(y=\cos x\), start by plotting the key points of the cosine function: \({(0,1), \left(\frac{\pi}{2}, 0\right), (\pi, -1),\left(\frac{3\pi}{2}, 0\right), (2\pi, 1)}\). These points represent the local maxima, local minima, and zeros of the function. Because \(\cos x\) is a periodic function with a period of \(2\pi\), the pattern of these points repeats every \(2\pi\) units on the x-axis. Sketch the graph by connecting the key points with a smooth curve, and extend the graph to the left and right as desired.
02

Sketch \(y=\cos^{-1} x\)

Now, let's sketch the graph of \(y=\cos^{-1} x\). The inverse cosine function is defined for values of \(x\) between -1 and 1, so mark these points on the x-axis. Recall that the outputs of \(y=\cos^{-1} x\) lie in the interval \([0, \pi]\). We have the following key points for the arccosine function: \({(-1, \pi), (0, \frac{\pi}{2}), (1, 0)}\). Connect the key points with a smooth curve to sketch the graph of \(y=\cos^{-1}x\). The graph should be increasing, and should not extend beyond the x-interval \([-1,1]\).
03

Reflections

To better understand the relationship between the graphs of \(y=\cos x\) and \(y=\cos^{-1} x\), notice that the graph of \(y=\cos^{-1} x\) is a reflection of the graph of \(y=\cos x\) over the line \(y = x\). This reflection property holds for all inverse functions.
04

Label the graphs

Finally, to complete the sketch, label the graphs of \(y=\cos x\) and \(y=\cos^{-1} x\) appropriately. Clearly marking each curve with its corresponding function will make it easier to compare and analyze the graphs.

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