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Chapter 4: Problem 61

Think About It 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 \(h(t)=f(t) g(t)=\sin t \cdot \cos t\) is an odd function because \(h(-t)= -h(t)\).

Step by step solution

01

Definition of Odd and Even Functions

An odd function satisfies \(f(-x)=-f(x)\) and an even function satisfies \(f(-x)=f(x)\). These definitions are central to understanding how odd and even functions interact.
02

Applying Definitions to \(f(t)\) and \(g(t)\)

By our definition, \(f(t)=\sin t\) is an odd function because \(\sin(-t)=-\sin t\) and \(g(t)=\cos t\) is an even function because \(\cos(-t)=\cos t\).
03

Determining the Nature of \(h(t)\)

Given function \(h(t)=f(t) g(t)=\sin t \cdot \cos t\), substituting \(-t\) for \(t\) gives \(h(-t)= f(-t)g(-t)=(-\sin t) \cdot (\cos t) = -h(t)\). So, \(h(t)\) is an odd function

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