If √7 sin θ = 3 cos θ, 0° < θ < 90°, then the value of \(\rm \sqrt{9 cosec^2\theta+7\sec^2\theta}\) is

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AAI Junior Assistant (Fire Service) Official Paper (Held On: 15 Nov 2022 Shift 1)
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  1. 4√2
  2. 4
  3. 2√2
  4. 2

Answer (Detailed Solution Below)

Option 1 : 4√2
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Detailed Solution

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Given:

√7 sin θ = 3 cos θ

Formula used:

cosec θ = 1/sin θ

sec θ = 1/cos θ

sin2θ + cos2θ = 1

Calculation:

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√7 sin θ = 3 cos θ

⇒ sin θ / cos θ = 3 / √7

⇒ tan θ = 3 / √7

⇒ sin2θ = 9 / (9 + 7) = 9/16

⇒ sin θ = 3/4

⇒ cosec θ = 4/3

cos2θ = 7 / 16

⇒ cos θ = √7 / 4

⇒ sec θ = 4 / √7

Now, √(9 × cosec2θ + 7 × sec2θ)

⇒ √(9 × (16/9) + 7 × (16/7))

⇒ √(16 + 16)

⇒ √32

⇒ 4√2

∴ The correct answer is 4√2.

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