sin5x=cos2x ⇒cos2x=cos(π2 5x) ⇒ 2x=2nπ±(π2 5x) Taking positive sign, 2x=2nπ+π2 5x ⇒ 7x=2nπ+π2 ⇒ x=(4n+1)π14 Taking negative sign, ...

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Trigonometric Ratios of Compound Angles

Solve: sin5x...

Question

Solve: $sin 5 x = cos 2 x$

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Solution

$sin 5 x = cos 2 x$
$\Rightarrow cos 2 x = cos (\frac{π}{2} - 5 x)$
$\Rightarrow 2 x = 2 n π \pm (\frac{π}{2} - 5 x)$
Taking positive sign,
$2 x = 2 n π + \frac{π}{2} - 5 x$
$\Rightarrow 7 x = 2 n π + \frac{π}{2}$
$\Rightarrow x = (4 n + 1) \frac{π}{14}$
Taking negative sign,
$2 x = 2 n π - \frac{π}{2} + 5 x$
$\Rightarrow - 3 x = 2 n π - \frac{π}{2}$
$\Rightarrow - 3 x = (4 n - 1) \frac{π}{2}$
$\Rightarrow x = - (4 n - 1) \frac{π}{6}$
Hence, $x = (4 n + 1) \frac{π}{14}$ or $x = - (4 n - 1) \frac{π}{6}$ where $n = 0, \pm 1, \pm 2, . . .$

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