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Chapter 4: Problem 70

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\sin x-\cos \left(x+\frac{\pi}{2}\right), \quad g(x)=2 \sin x$$

Short Answer

Expert verified
The functions \(f(x) = \sin x - \cos \left(x+\frac{\pi}{2}\right)\) and \(g(x) = 2\sin x\) are identical.

Step by step solution

01

Simplify the function \(f(x)\)

Simplify the function \(f(x)\) by using the trigonometric identity that \(\cos(x + \frac{\pi}{2}) = -\sin(x)\). This results in \(f(x) = \sin(x) - (-\sin(x)) = 2\sin(x)\).
02

Graph the functions

Both functions, \(f(x)\) and \(g(x)\) should be graphed on the same coordinate system. Since it has been established that both functions are equivalent, the two graphs will overlap, appearing as a single line.
03

Make a conjecture about the relationship between the two functions

Having graphed the two functions, and seeing that they overlap, the conjecture is that the functions \(f(x)\) and \(g(x)\) are identical.

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