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Chapter 4: Problem 69

Because \(f(t)=\sin t\) is an odd function and \(g(t)=\cos t\) is an even function, what can be said about the function \(h(t)=f(t) g(t) ?\)

Short Answer

Expert verified
The function \(h(t)=f(t)g(t)\) is an odd function.

Step by step solution

01

Recall the properties of even and odd functions

An even function follows the rule \(f(-x) = f(x)\). An odd function follows the rule \(f(-x) = -f(x)\).
02

Determine the nature of the newly formed function

Calculate \(h(-t)\), which is equal to \(f(-t)g(-t)\). Since \(f(t)= \sin t\) is an odd function, substitute \(f(-t)\) with \(-f(t)\). And since \(g(t) = \cos t\) is an even function, substitute \(g(-t)\) with \(g(t)\). This gives us \(h(-t) = -f(t)g(t)\), which is equal to \(-h(t)\). Thus, we can conclude that \(h(t)\) is an odd function.
03

Conclusion

The product of an odd function and an even function is an odd function. Therefore, \(h(t)=f(t)g(t)\) is an odd function.

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Most popular questions from this chapter

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ f(x)=\tan x $$

Fill in the blanks. $$ \begin{array}{ll} \text { Function } & \text { Alternative Notation } & \text { Domain } & \text { Range } \end{array} $$ _________ $$ y=\cos ^{-1} x \quad-1 \leq x \leq 1 $$__________

Sketch the graph of the function. Include two full periods. $$ y=\tan (x+\pi) $$

Use a graphing utility to graph the function. Include two full periods. $$ y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right) $$

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