sin 3x lt; sin x sin 3x sin x lt; 0 Put sin 3x=3 sin x 4 sin^3x 3 sin x 4 sin^3x sin x lt; 0 2 sin x 4 sin^3x lt; 0 2 ...

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Domain and Range of Trigonometric Ratios

Solve sin 3...

Question

Solve $sin 3 x < sin x$

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Solution

$sin 3 x < sin x$
$sin 3 x - sin x < 0$
Put $sin 3 x = 3 sin x - 4 {sin}^{3} x$
$3 sin x - 4 {sin}^{3} x - sin x < 0$
$2 sin x - 4 {sin}^{3} x < 0$
$2 sin x (1 - 2 {sin}^{2} x) < 0$
$sin x cos 2 x < 0$ $[cos 2 x = 1 - 2 {sin}^{2} x]$
$sin x \to + v e$ $cos 2 x \to - v e$
$x ϵ (\frac{π}{4}, \frac{π}{2}] \cup [\frac{π}{2}, \frac{3 π}{4}) = (\frac{π}{4}, \frac{3 π}{4})$
$x \to - v e$ $cos 2 x \to + v e$
$x ϵ (π, \frac{5 π}{4}) \cup (\frac{7 π}{4}, 2 π)$
General solution will be $x ϵ (x π + \frac{π}{4} + n π + \frac{3 π}{4}) \cup (n π + π, n π + \frac{5 π}{4}) \cup (n π + \frac{7 π}{4} + n π + 2 π)$
Where n is even number.

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