91Ó°ÊÓ

We have given that following function :-

$f\left(\mathrm{θ}\right)=csc\mathrm{θ}$and $f\left(a\right)=2$.

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 :-

$f\left(\mathrm{θ}\right)=csc\mathrm{θ}$and $f\left(a\right)=2$.

We know that :-

$csc(-\mathrm{θ})=-csc\mathrm{θ}$.

Now put $\mathrm{θ}=-a$, in $f\left(\mathrm{θ}\right)=csc\mathrm{θ}$, then we have :-

$$\begin{gathered} f(-a)=csc(-a) \\ ⇒f(-a)=-csc\left(a\right) \\ ⇒f(-a)=-f\left(a\right) \end{gathered}$$

Now put $f\left(a\right)=2$, then we have :-

$f(-a)=-2$.

This is the required value.

We have given that :-

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

We know that cosecant function is of period $2\mathrm{Ï€}$.

This gives us :-

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

Now :-

role="math" localid="1647130898644" $$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=csc\left(a\right)+csc(a+2\mathrm{π})+csc(a+4\mathrm{π}) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=csc\left(a\right)+csc\left(a\right)+csc\left(a\right) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3csc\left(a\right) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3f\left(a\right) \end{gathered}$$

Put $f\left(a\right)=2$, then we have :-

$$\begin{gathered} f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=3\left(2\right) \\ ⇒f\left(a\right)+f(a+2\mathrm{π})+f(a+4\mathrm{π})=6 \end{gathered}$$

This is the required value.