91Ó°ÊÓ

We have given that following function :-

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

We have to find the value of $f(-a)$and value of$f\left(a\right)+f(a+2\mathrm{Ï€})+f(a-2\mathrm{Ï€})$.

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

We have given that :-

$f\left(\mathrm{θ}\right)=\cos \mathrm{θ}$.

Put $\mathrm{θ}=-a$, then we have :-

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

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

$\cos \left(-\mathrm{θ}\right)=\cos \mathrm{θ}$.

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.

Given that :-

$f\left(\mathrm{θ}\right)=\cos \mathrm{θ}$.

We know that cosine function is periodic for period $2\mathrm{Ï€}$.

That is :-

$\cos (\mathrm{θ}+2\mathrm{π}k)=\cos \mathrm{θ}$for any integer $k$.

So that we have :-

$$\begin{gathered} f\left(\mathrm{θ}\right)+f(\mathrm{θ}+2\mathrm{π})+f(\mathrm{θ}-2\mathrm{π})=\cos \mathrm{θ}+\cos (\mathrm{θ}+2\mathrm{π})+\cos (\mathrm{θ}-2\mathrm{π}) \\ ⇒f\left(\mathrm{θ}\right)+f(\mathrm{θ}+2\mathrm{π})+f(\mathrm{θ}-2\mathrm{π})=\cos \mathrm{θ}+\cos \mathrm{θ}+\cos \mathrm{θ} \\ ⇒f\left(\mathrm{θ}\right)+f(\mathrm{θ}+2\mathrm{π})+f(\mathrm{θ}-2\mathrm{π})=3\cos \mathrm{θ} \\ ⇒f\left(\mathrm{θ}\right)+f(\mathrm{θ}+2\mathrm{π})+f(\mathrm{θ}-2\mathrm{π})=3f\left(\mathrm{θ}\right). \end{gathered}$$

Now take $a$in place of $\mathrm{θ}$, then we have :-

role="math" localid="1646872305545" $f\left(a\right)+f(a+2\mathrm{Ï€})+f(a-2\mathrm{Ï€})=3f\left(a\right)$.

We have :-

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

By putting this value we have :-

$$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a-2\mathrm{π})=3×\frac{1}{4} \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a-2\mathrm{π})=\frac{3}{4} \end{gathered}$$

This is the required value.