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Chapter 0: Problem 70

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false. The product \(y=(\sin x)(\cos x)\) is an odd function of \(x\).

Short Answer

Expert verified
The statement is true. The product \(y = (\sin x)(\cos x)\) is an odd function of \(x\), because when evaluating \(y(-x)\), we find that \(y(-x) = -(\sin(x)(\cos(x)))\), which is equal to \(-y(x)\). This satisfies the condition for odd functions, \(f(-x) = -f(x)\).

Step by step solution

01

Define the odd function

An odd function is a function that satisfies the following condition: \(f(-x) = -f(x)\) for all values of x in the domain of the function.
02

Identify the given function

We are given the function \(y = (\sin x)(\cos x)\).
03

Evaluate y(-x)

Substitute -x for x in the given function: \(y(-x) = (\sin(-x))(\cos(-x))\)
04

Apply trigonometric rules

Recall the rules of sine and cosine for negative angles: \[\sin(-x) = -\sin(x) \and \cos(-x) = \cos(x)\] Substitute these rules into y(-x): \[y(-x) = (-\sin(x))(\cos(x)) = -(\sin(x)(\cos(x)))\]
05

Compare y(-x) and -y(x)

We have found that: \[y(-x) = -(\sin(x)(\cos(x)))\] And: \[-y(x) = -(\sin(x)(\cos(x)))\] Both expressions are the same, so the function satisfies the condition for odd functions. Therefore, the given statement is true: The product \(y = (\sin x)(\cos x)\) is an odd function of \(x\).

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Most popular questions from this chapter

Let $$ f(x)=\left\\{\begin{array}{ll} 2 x-1 & \text { if } x<1 \\ \sqrt{x} & \text { if } 1 \leq x<4 \\ \frac{1}{2} x^{2}-6 & \text { if } x \geq 4 \end{array}\right. $$ Find \(f^{-1}(x)\), and state its domain.

Find the inverse of \(f .\) Then use a graphing utility to plot the graphs of \(f\) and \(f^{-1}\) using the same viewing window. $$ f(x)=1-\frac{1}{x} $$

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\sin x, \quad y=2 \sin \frac{x}{2}\)

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{x^{3}}{x^{3}+1} $$

a. Plot the graph of \(f(x)=\cos (\sin x)\). Is \(f\) odd or even? b. Verify your answer to part (a) analytically.
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