First slide

Trigonometric equations

Question

A set of value of x satisfying the equation $\cos^{2} (\frac{1}{2} p x) + \cos^{2} (\frac{1}{2} q x) = 1$ form an arithmetic progression with common difference

Difficult

A

$\frac{2 π}{p + q}$
B

$\frac{2 π}{p - q}$
C

$\frac{π}{p + q}$
D

$2 π$

Solution

$1 + \cos p x + 1 + \cos q x = 2$
$\cos (\frac{p + q}{2}) x \cos (\frac{p - q}{2}) x = 0$
$\cos (\frac{p + q}{2}) x = 0 \Rightarrow x = \frac{(2 n + 1) π}{p + q}, n = 0, \pm 1, \pm 2....$ which forms an A.P with common difference $\frac{2 π}{p + q}$ .
and $\cos (\frac{p - q}{2}) x = 0 \Rightarrow x = \frac{(2 n + 1) π}{p - q}, n = 0, \pm 1, \pm 2...$ which forms an A.P with common difference $\frac{2 π}{p - q}$

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