Let $\mathrm{S}=\{\theta \in(-2 \pi, 2 \pi): \cos \theta+1=\sqrt{3} \sin \theta\}$. Then $\sum_{\theta \in \mathrm{S}} \theta$ is equal to:

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Trigonometric Equations

Let

S = {θ \in (- 2 π, 2 π) : cos θ + 1 = 3 sin θ}

.

Then

\sum_{θ \in S} θ

is equal to:

Answer: (B)

Using

1 + cos θ = 2 cos^{2} \frac{θ}{2}

and

sin θ = 2 sin \frac{θ}{2} cos \frac{θ}{2}

, the equation

cos θ + 1 = 3 sin θ

becomes

2 cos^{2} \frac{θ}{2} = 23 sin \frac{θ}{2} cos \frac{θ}{2}

. Dividing by

2 cos \frac{θ}{2}

(the case

cos \frac{θ}{2} = 0

fails the original equation) gives

tan \frac{θ}{2} = \frac{1}{3}

. For

θ \in (- 2 π, 2 π)

\frac{θ}{2} \in (- π, π)

, so

\frac{θ}{2} = \frac{π}{6}

- \frac{5 π}{6}

, giving

θ = \frac{π}{3}

- \frac{5 π}{3}

. Their sum is

\frac{π}{3} - \frac{5 π}{3} = - \frac{4 π}{3}

JEE Main · 2026 · 06 Apr 2026 · Shift 1 · Single correct option