The number of solutions of the equation $s in x + s in 5 x = s in 3 x$ lying in the interval $[0, π]$ is

Answer (Detailed Solution Below)View answer ▾

Answer: Option B — 6

Detailed Solution

$s in x + s in 5 x = 2 s in 3 x cos 2 x$ , so the equation becomes $s in 3 x (2 cos 2 x - 1) = 0$ .

$s in 3 x = 0$ gives $x = 0, \frac{π}{3}, \frac{2 π}{3}, π$ : 4 values.

$cos 2 x = \frac{1}{2}$ gives $2 x = \frac{π}{3}, \frac{5 π}{3}$ , so $x = \frac{π}{6}, \frac{5 π}{6}$ : 2 values. Total 6.

✓ Check

x = \frac{π}{6}

s in \frac{π}{6} + s in \frac{5 π}{6} = \frac{1}{2} + \frac{1}{2} = 1 = s in \frac{π}{2}

⚠ Common trap

(D) 2 comes from cancelling

s in 3 x

from both sides, which loses the four roots of

s in 3 x = 0

Quick tip

$sin 3 x (2 cos 2 x - 1) = 0$ : 4 + 2 = 6 roots

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Trigonometric Equations — Preparation

This question tests your understanding of Trigonometric Equations, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A trigonometric equation is an equation involving trigonometric functions of an unknown angle, such as 2 sin x = 1. This topic covers finding principal solutions (usually between 0 and 2 pi) and general solutions, which include all possible angles. The key general solutions are: sin x = sin a gives x = n pi + (-1)^n a, cos x = cos a gives x = 2n pi +/- a, and tan x = tan a gives x = n pi + a, where n is any integer. Exam questions involve reducing an equation to one of these standard forms, often by factorising or using identities, and then counting the number of solutions in a given interval. Always check that solutions do not make any term in the original equation undefined. Practise the trigonometric equations questions below, each with a step-by-step solution.

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