# Solve the Following Equation: Cos X + Cos 2 X + Cos 3 X = 0 - Mathematics

Sum

Solve the following equation:

$\cos x + \cos 2x + \cos 3x = 0$

#### Solution

$\cos x + \cos 2x + \cos 3x = 0$

Now,

$(\cos x + \cos3x) + \cos2x = 0$
$\Rightarrow 2 \cos \left( \frac{4x}{2} \right) \cos \left( \frac{2x}{2} \right) + \cos2x = 0$
$\Rightarrow 2 \cos2x \cos x + \cos2x = 0$
$\Rightarrow \cos2x ( 2 \cos x + 1) = 0$

$\Rightarrow \cos 2x = 0$ or,
$2 \cos x + 1 = 0$
$\Rightarrow \cos 2x = \cos \frac{\pi}{2}$ or
$\cos x = - \frac{1}{2} = \cos \frac{2\pi}{3}$
$\Rightarrow 2x = (2n + 1) \frac{\pi}{2}$,
$n \in Z$ or

$x = 2m\pi \pm \frac{2\pi}{3}, m \in Z$

$\Rightarrow x = (2n + 1)\frac{\pi}{4}, n \in Z$
$x = 2m\pi \pm \frac{2\pi}{3}, m \in Z$
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Exercise 11.1 | Q 4.1 | Page 22