First slide

Trigonometric equations

Question

$I f 0 \leq x < 2 π, t h e n t h e n u m b e r o f r e a l v a l u e s o f x, w h i c h s a t i s f y t h e e q u a t i o n \cos x + \cos 2 x + \cos 3 x + \cos 4 x = 0, i s$

Easy

A

5
B

7
C

9
D

3

Solution

$\cos 4 x + \cos x + \cos 3 x + \cos 2 x = 0 \Rightarrow 2 \cos \frac{5 x}{2} \cos \frac{3 x}{2} + 2 \cos \frac{5 x}{2} \cos \frac{x}{2} = 0 (b y t r a n s f o r m a t i o n s)$
$\Rightarrow \cos \frac{5 x}{2} [\cos \frac{3 x}{2} + \cos \frac{x}{2}] = 0 \Rightarrow \cos \frac{5 x}{2} [2 \cos x \cos \frac{x}{2}] = 0 \Rightarrow ∴ x = \frac{π}{5}, \frac{π}{2}, \frac{3 π}{5}, π, \frac{7 π}{5}, \frac{3 π}{2}, \frac{9 π}{5}$

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