Q 113 - Precalculus Enhanced with Graphing Utilities

Question

In problems use the periodic and even–odd properties. 

If $f\left(\theta \right)=\sin \left(\theta \right)$ and $f\left(a\right)=\frac{1}{3}$, find the exact value of :

(a) $f(-a)$

(b) $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$.

Step-by-Step Solution

Answer

(a) The value of $f(-a)$ is $\frac{1}{3}$.

(b) The value of $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$ is $1$.

Step 1. Given Information

We have given that following function :-

$f\left(\theta \right)=\sin \left(\theta \right)$ and $f\left(a\right)=\frac{1}{3}$.

We have to find the value of$f(-a)$  and value of $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$.

To find value of $f(-a)$ we will use even-odd properties and to find the value of $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$ we will use periodic properties.

Step 2. Part (a). To find value of f ( - a )

We have given that :-

$f\left(\theta \right)=\sin \left(\theta \right)$.

Put $\theta =-a$, then we have :-

$f(-a)=\sin \left(-a\right)$.

We know that sine function is an odd function, that is :-

$\sin \left(-\theta \right)=-\sin \left(\theta \right)$.

So that $\sin \left(-a\right)=-\sin \left(a\right)$.

This gives us :-

$$\begin{gathered} f(-a)=-\sin \left(a\right) \\ or f(-a)=-f\left(a\right) \end{gathered}$$

We have given that $f\left(a\right)=\frac{1}{3}$.

By putting this value we have :-

$f(-a)=-\frac{1}{3}$.

This is the required value.

Step 3. Part (b) To find value of f ( a ) + f ( a + 2 π ) + f ( a + 4 π )

Given that :-

$f\left(\theta \right)=\sin \left(\theta \right)$.

We know that sine function is periodic for period $2\pi$.

That is:-

$\sin \left(\theta +2\pi k\right)=\sin \left(\theta \right)$, for any integer $k$.

So that we have :-

$$\begin{gathered} f\left(\theta \right)+f(\theta +2\pi )+f(\theta +4\pi )=\sin \left(\theta \right)+\sin \left(\theta +2\pi \right)+\sin \left(\theta +4\pi \right) \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta +4\pi )=\sin \left(\theta \right)+\sin \left(\theta \right)+\sin \left(\theta \right) \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta +4\pi )=3\sin \left(\theta \right) \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta +4\pi )=3f\left(\theta \right) \end{gathered}$$

Now take $a$ in place of $\theta$, then we have :-

$f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3f\left(a\right)$.

We have $f\left(a\right)=\frac{1}{3}$.

By putting this value we have :- 

$$\begin{gathered} f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3\times \frac{1}{3} \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=1 \end{gathered}$$

This is the required value.