91Ó°ÊÓ

We have given that following function :-

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

We have to find the value of$f(-a)$ and value of $f\left(a\right)+f(a+2\mathrm{Ï€})+f(a+4\mathrm{Ï€})$.

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

We have given that :-

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

Put localid="1646838403718" $\mathrm{θ}=-a$, then we have :-

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

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

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

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

This gives us :-

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

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

By putting this value we have :-

localid="1646838673711" $f(-a)=-\frac{1}{3}$.

This is the required value.

Given that :-

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

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

That is:-

$\sin \left(\mathrm{θ}+2\mathrm{π}k\right)=\sin \left(\mathrm{θ}\right)$, for any integer $k$.

So that we have :-

Now take localid="1646839312516" $a$in place of $\mathrm{θ}$, then we have :-

$f\left(a\right)+f(a+2\mathrm{Ï€})+f(a+4\mathrm{Ï€})=3f\left(a\right)$.

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

By putting this value we have :-

localid="1646871511245" $$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3×\frac{1}{3} \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=1 \end{gathered}$$

This is the required value.