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NDA 2022 Maths · Mathematics · Trigonometric Equations · medium

The value of θ∈[0,π] satisfying sin θ+√3cos θ=2 is

Single correct option

The value of θ∈[0,π]\theta\in[0,\pi] satisfying sin⁡θ+3cos⁡θ=2\sin\theta+\sqrt3\cos\theta=2 is
  1. A. π6\dfrac\pi6
  2. B. π3\dfrac\pi3
  3. C. π2\dfrac\pi2
  4. D. 2π3\dfrac{2\pi}3
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Answer: (A) π6\dfrac\pi6

sin⁡θ+3cos⁡θ=2sin⁡(θ+π3)=2\sin\theta+\sqrt3\cos\theta=2\sin\left(\theta+\frac\pi3\right)=2 gives θ+π3=π2\theta+\frac\pi3=\frac\pi2, so θ=π6\theta=\frac\pi6. The maximum of the left side is 22, so this is the only solution.
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