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NDA 2025 Maths · Mathematics · Trigonometric Equations · medium

The number of values of θ in [0,2π) satisfying sin θ+cos θ=1 is

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The number of values of θ\theta in [0,2π)[0,2\pi) satisfying sin⁡θ+cos⁡θ=1\sin\theta+\cos\theta=1 is
  1. A. 22
  2. B. 11
  3. C. 33
  4. D. 44
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Answer: (A) 22

Write 2sin⁡(θ+π4)=1\sqrt2\sin(\theta+\frac\pi4)=1, so sin⁡(θ+π4)=12\sin(\theta+\frac\pi4)=\frac1{\sqrt2} with θ+π4∈[π4,9π4)\theta+\frac\pi4\in[\frac\pi4,\frac{9\pi}4). This gives π4\frac\pi4 and 3π4\frac{3\pi}4, i.e. θ=0\theta=0 and θ=π2\theta=\frac\pi2. Two solutions.
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