Question

Use the periodic and even–odd properties.

If $f\left(\theta \right)=\cos \theta$and $f\left(a\right)=\frac{1}{4}$, find the exact value of :

(a) $f(-a)$

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

Short answer

(a) The value of $f(-a)$is$\frac{1}{4}$.

(b) The value of $f\left(a\right)+f(a+2\pi )+f(a-2\pi )$is$\frac{3}{4}$.

Step-by-step solution

Step 1. Given Information 

We have given that following function :-

$f\left(\theta \right)=\cos \theta$and $f\left(a\right)=\frac{1}{4}$.

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

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

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

We have given that :-

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

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

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

We know that cosine function is an even function, that is :

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

So that :-

role="math" localid="1646871733428" $\cos \left(-a\right)=\cos \left(a\right)$.

This gives us :-

role="math" localid="1646871721683" $$\begin{gathered} f\left(-a\right)=\cos \left(a\right) \\ orf(-a)=f\left(a\right) \end{gathered}$$

We have given that :-

$f\left(a\right)=\frac{1}{4}$.

By putting this value we have :-

$f(-a)=\frac{1}{4}$.

This is the required value.

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

Given that :-

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

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

That is :-

$\cos (\theta +2\pi k)=\cos \theta$for any integer $k$.

So that we have :-

$$\begin{gathered} f\left(\theta \right)+f(\theta +2\pi )+f(\theta -2\pi )=\cos \theta +\cos (\theta +2\pi )+\cos (\theta -2\pi ) \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta -2\pi )=\cos \theta +\cos \theta +\cos \theta \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta -2\pi )=3\cos \theta \\ \Rightarrow f\left(\theta \right)+f(\theta +2\pi )+f(\theta -2\pi )=3f\left(\theta \right). \end{gathered}$$

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

role="math" localid="1646872305545" $f\left(a\right)+f(a+2\pi )+f(a-2\pi )=3f\left(a\right)$.

We have :-

$f\left(a\right)=\frac{1}{4}$.

By putting this value we have :-

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

This is the required value.