Question

Use the periodic and even–odd properties.

If$f\left(\theta \right)=sec\theta$and $f\left(a\right)=-4$, find the exact value of :

(a) $f(-a)$

(b) $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$.

Short answer

(a) The value of $f(-a)$is$-4$.

(b) The value of $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$is$-12$.

Step-by-step solution

Step 1. Given Information  

We have given that following function :-

$f\left(\theta \right)=sec\theta$and role="math" $f\left(a\right)=-4$.

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 $f\left(a\right)+f(a+2\pi )+f(a+4\pi )$we will use periodic properties.

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

We have given that :-

$f\left(\theta \right)=sec\theta$and $f\left(a\right)=-4$.

We know that :-

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

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

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

Put $f\left(a\right)=-4$, then we have :-

$f(-a)=-4$.

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 :-

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

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

This gives us :-

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

Now :-

localid="1647095654962" $$\begin{gathered} f\left(a\right)+f(a+2\pi )+f(a+4\pi )=sec\left(a\right)+sec(a+2\pi )+sec(a+4\pi ) \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=sec\left(a\right)+sec\left(a\right)+sec\left(a\right) \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3sec\left(a\right) \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3f\left(a\right) \end{gathered}$$

Put $f\left(a\right)=-4$, then we have :-

$$\begin{gathered} f\left(a\right)+f(a+2\pi )+f(a+4\pi )=3(-4) \\ \Rightarrow f\left(a\right)+f(a+2\pi )+f(a+4\pi )=-12 \end{gathered}$$

This is the required value.