Question

Let $f$ be an odd function defined on the real numbers such that $ f(x) = 3 \, \sin \, x + 4 \, \cos \, x,$ for $x \geq 0,$ then $f(x)$ for $x < 0$ is :

• A. $3 \, \sin \, x - 4 \, \cos \, x $
• B. $-3 \, \sin \, x + 4 \, \cos \, x $
• C. $ - 3 \, \sin \, x - 4 \, \cos \, x $
• D. $3 \, \sin \, x + 4 \, \cos \, x $

Answer 1

Answer: (A)

Given, $f(x)=3 \sin x+4 \cos\, x$
When, $X<\,0$
$\therefore \,-X>\,0$
$\therefore \, f(-x)=3 \sin (-x)+4 \cos (-x)\,...(i)$
But, $f(x)$ is odd function
$\therefore \, f(-X) =-f(x)$, when $x<\,0$
$\Rightarrow \, f(x) =-f(-X) $
$=-[3 \sin (-x)+4 \cos (-x)]$ [From Eq. (i)]
$=3 \sin x-4 \cos\, x$