If sinθ + 2cosθ = 1, find the value of 2sinθ - cosθ.


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Answer:

2

Step by Step Explanation:
  1. It is given that
    sinθ + 2cosθ = 1
    ⇒ (sinθ + 2cosθ)2 = 12 ....... [On squaring both sides]
  2. Now, add (2sinθ - cosθ)2 to both sides of equation
    ⇒ (sinθ + 2cosθ)2 + (2sinθ - cosθ)2 = 12 + (2sinθ - cosθ)2
    ⇒ sin2θ + 4cos2θ + 4sinθcosθ + 4sin2θ + cos2θ - 4sinθcosθ = 1 + (2sinθ - cosθ)2
    ⇒ 5sin2θ + 5cos2θ + 4sinθcosθ - 4sinθcosθ = 1 + (2sinθ - cosθ)2
    ⇒ 5(sin2θ + cos2θ) = 1 + (2sinθ - cosθ)2
    ⇒ 5 = 1 + (2sinθ - cosθ)2
    ⇒ (2sinθ - cosθ)2 = 5 - 1
    ⇒ (2sinθ - cosθ)2 = 4
    ⇒ 2sinθ - cosθ = ± 2
  3. As sinθ + 2cosθ = 1, 2sinθ - cosθ will always be positive.
    Therefore, 2sinθ - cosθ = 2

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