Chapter 14.6, Problem 49PPSE

Solution

a.

To identify: The trigonometric function that are even and odd if a function $f\left(-x\right)=f\left(x\right)$ is an even function and a function $f\left(-x\right)=-f\left(x\right)$ is odd.

The cosine function is even and the sine and tangent functions are odd.

Given information:

An even function is $f\left(-x\right)=f\left(x\right)$ and an odd function is $f\left(-x\right)=-f\left(x\right)$ .

Explanation:

The given functions are: An even function $f\left(-x\right)=f\left(x\right)$ and an odd function $f\left(-x\right)=-f\left(x\right)$

The trigonometric cosine function is even and the sine and tangent functions are odd.

  $\begin{array}{l}\sin \left(-\theta \right)=-\sin \theta \\ \tan \left(-\theta \right)=-\tan \theta \end{array}$

On the other hand, $\cos \left(-\theta \right)=\cos \theta$ .

Therefore, the cosine function is even and the sine and tangent functions are odd.

b.

To state: If all the functions either even or odd also give a counter example if possible.

No, all the functions are not either odd or even.

Given information:

An even function is $f\left(-x\right)=f\left(x\right)$ and an odd function is $f\left(-x\right)=-f\left(x\right)$ .

Explanation:

No, all functions are not even or add.

The reason being $\cos \left(-x\right)=\cos x$ is an even function.

Whereas $\sin \left(-x\right)=-\sin x$ is an odd function.

Therefore, all the functions cannot be defined either odd or even.