JEE Main 2026 Apr 6 Shift 1, Mathematics Q14: Trigonometric Equations

JEE Main2026Apr 6, Shift 1Mathematics

Let $S = {θ \in (- 2 π, 2 π) : cos θ + 1 = 3 sin θ}$ . Then $θ \in S \sum θ$ is equal to:

Solution

Rewrite as $3 sin θ - cos θ = 1$ . Using the $R$ -method, $3 sin θ - cos θ = 2 sin (θ - \frac{π}{6})$ , so $sin (θ - \frac{π}{6}) = \frac{1}{2}$ .

Hence $θ - \frac{π}{6} = \frac{π}{6} + 2 nπ$ or $θ - \frac{π}{6} = \frac{5 π}{6} + 2 nπ$ , giving $θ = \frac{π}{3} + 2 nπ$ or $θ = π + 2 nπ$ .

Intersecting with $θ \in (- 2 π, 2 π)$ : $S = {\frac{π}{3}, - \frac{5 π}{3}, π, - π}$ .

Sum $= \frac{π}{3} - \frac{5 π}{3} + π - π = - \frac{4 π}{3}$ .

Concept behind this question

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