Question

Use the periodic and even–odd properties :

If $f\left(\theta \right)=cot\theta$and $f\left(a\right)=-3$, find the exact value of :

(a)$f(-a)$

(b) $f\left(a\right)+f(a+\pi )+f(a+4\pi )$.

Short answer

(a) The value of $f(-a)$is$3$.

(b) The value of $f\left(a\right)+f(a+\pi )+f(a+4\pi )$is$-9$.

Step-by-step solution

Step 1. Given Information 

We have given that following function :-

$f\left(\theta \right)=cot\theta$and $f\left(a\right)=-3$.

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

To find value of $f(-a)$we will use even-odd properties and to find the value of $f\left(a\right)+f(a+\pi )+f(a+4\pi )$we will use periodic properties.

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

We have given that :-

$f\left(\theta \right)=cot\theta$and $f\left(a\right)=-3$.

We know that :-

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

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

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

Put $f\left(a\right)=-3$, then we have :-

$$\begin{gathered} f(-a)=-(-3) \\ \Rightarrow f(-a)=3 \end{gathered}$$

This is the required value.

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

We have :-

$f\left(\theta \right)=cot\theta$

We know that cotangent function is periodic of period $\pi$.

This gives us :-

$cot(\theta +\pi k)=cot\theta$.

Now :-

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

Put $f\left(a\right)=-3$, then we have :-

$$\begin{gathered} f\left(a\right)+f(a+\pi )+f(a+4\pi )=3(-3) \\ \Rightarrow f\left(a\right)+f(a+\pi )+f(a+4\pi )=-9 \end{gathered}$$

This is the required value.