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Chapter 4: Problem 97

Graph \(f\) and \(g\) in the same coordinate plane. Include two full periods. Make a conjecture about the functions. $$f(x)=\sin x, \quad g(x)=\cos \left(x-\frac{\pi}{2}\right)$$

Short Answer

Expert verified
The functions \(f(x) = \sin x\) and \(g(x) = \cos (x - \frac{\pi}{2})\) are equivalent. This is evident because their graphs overlap exactly for the range considered (0 to \(4\pi\)).

Step by step solution

01

Graph \(f(x) = \sin x\)

Start by graphing \(f(x) = \sin x\) on the coordinate plane over two periods, that is, from \(x = 0\) to \(x = 4\pi\). The sine function has a maximum point of (0, 1), crosses the x-axis at \(\pi\) and \(2\pi\), and has a minimum point at \((3\pi/2, -1)\). It then crosses the x-axis again at \(2\pi\), and these patterns repeat for the second period.
02

Graph \(g(x) = \cos (x - \frac{\pi}{2})\)

Next, graph \(g(x) = \cos (x - \frac{\pi}{2})\) on the same coordinate plane. This function is a horizontal shift of the cosine function, moved to the right by \(\pi/2\). It starts at the point \((\pi/2, 1)\), crosses the x-axis at \((\pi, 0)\) and \((3\pi/2, 0)\), and has a minimum point at \((2\pi, -1)\). This pattern also repeats for the second period.
03

Make a Conjecture

Once both functions are graphed, look for patterns or relationships. It can be observed that the graph of \(g(x)\) overlaps exactly with that of \(f(x)\). That means \(f(x) = g(x)\), or in other words, \(\sin x = \cos (x - \frac{\pi}{2})\) for all values of \(x\) in this scenario.

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