First slide

Trigonometric equations

Question

The number of solutions of $\sum_{r = 1}^{5} \cos rx = 5$ in the interval $[0, 2 π]$ is

Moderate

A

0
B

2
C

5
D

10

Solution

$\begin{array}{l} \sum_{r = 1}^{5} \cos rx = 5 \\ \Rightarrow \cos x + \cos 2 x + \cos 3 x + \cos 4 x + \cos 5 x = 5 \end{array}$
which is possible only when cosx= cos 2x= cos3x= cos 4x= cos5x = 1 and is satisfied by x = 0 and x = $2 π$

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