Question

sin(x) is an odd function and cos(x) is an even function.
a) Define f(x) as a modification of sin(x) so that f is even.
b) Define g(x) as a modification of cos(x) so that g is odd.

Answer 1

Task:

$\sin(x)$ is an odd function and $\cos(x)$ is an even function.

a) Define $f(x)$ as a modification of $\sin(x)$ so that $f$ is even.

b) Define $g(x)$ as a modification of $\cos(x)$ so that $g$ is odd.

Solution:

Let's build graph both $(\sin(x)$ and $\cos(x))$ functions on one plot:

(a)

On the plot above we see that $\sin(x)$ shifted by $\frac{\pi}{2}$ to the left, it will be superimposed on the cosine graph.

So $\sin\left(x + \frac{\pi}{2}\right) = \cos(x)$ and $\cos(x)$ is an even function. We get:

$$f(x) = \sin\left(x + \frac{\pi}{2}\right)$$

(b)

On the plot above we see that $\cos(x)$ shifted by $\frac{\pi}{2}$ to the left, looks so:

And

$$- \cos \left(x + \frac {\pi}{2}\right) = \cos \left(- \left(x + \frac {\pi}{2}\right)\right)$$

So $\cos \left(x + \frac{\pi}{2}\right)$ is an odd function. We get:

$$\mathrm {g} (\mathrm {x}) = \cos \left(x + \frac {\pi}{2}\right)$$

Answer: (a) $f(x) = \sin \left(x + \frac{\pi}{2}\right)$ and (b) $g(x) = \cos \left(x + \frac{\pi}{2}\right)$ .