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If \(\sec θ + \cos θ = \frac{5}{2}\), where 0 ≤ θ ≤ 90°, then what is the value of sin2 θ 
1. \(\frac{1}{4}\)
2. \(\frac{1}{2}\)
3. \(\frac{3}{4}\)
4. 1

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Correct Answer - Option 3 : \(\frac{3}{4}\)

Formula used:

  • cos θ = 1/sec θ 
  • sin2 θ + cos2 θ = 1

 

Calculation:

We have,

⇒ sec θ + cos θ = 5/2

⇒ 1/cos θ  + cos θ = 5/2

⇒ (1 + cos2 θ)/cos θ = 5/2

⇒ 2 + 2cos2 θ = 5cosθ

⇒ 2cos2θ - 5cosθ + 2 = 0

⇒ 2 cos2θ - 4cos θ - cos θ  + 2 = 0

⇒ 2cos θ(cos θ - 2) -1(cos θ - 2) = 0

⇒ (2cos θ - 1)(cos θ - 2) = 0

⇒ cosθ = 1/2

⇒ cos2 θ = 1/4

We know that,

⇒ sin2 θ + cos2 θ = 1

∴  sin2 θ = 1 - 1/4 = 3/4

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