Question

Use the periodic and even–odd properties.

If $f\left(\theta \right)=csc\theta$and $f\left(a\right)=2$, find the exact value of :

(a) $f(-a)$

(b) localid="1647096128554" $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$.

Short answer

(a) The value of $f(-a)$is$-2$.

(b) The value of localid="1647095937703" $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$is $6$.

Step-by-step solution

Step 1. Given Information   

We have given that following function :-

$f\left(\theta \right)=csc\theta$and $f\left(a\right)=2$.

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

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 :-

$f\left(\theta \right)=csc\theta$and $f\left(a\right)=2$.

We know that :-

$csc(-\theta )=-csc\theta$.

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

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

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

$f(-a)=-2$.

This is the required value.

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

We have given that :-

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

We know that cosecant function is of period $2\pi$.

This gives us :-

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

Now :-

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

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

$$\begin{gathered} f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3\left(2\right) \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=6 \end{gathered}$$

This is the required value.