If 3 sin θ + 5 cos θ = 5, then the value of

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If 3 sin θ + 5 cos θ = 5, then the value of

Home » Aptitude » Trigonometry » Question

If 3 sin θ + 5 cos θ = 5, then the value of 5 sin θ – 3 cos q will be

1. ± 3
2. ± 5
3. ± 2
4. ± 1

Correct Option: A

3 sin θ + 5 cos θ = 5 --- (i)
5 sin θ – 3 cos θ = x (let)-- (ii)
On squaring and adding both the equations,
(3sin θ + 5 cos θ) 2 + (5 sin θ – 3 cos θ )² = 5² + x²
⇒ 9 sin²θ + 2² cos²θ + 30 sin θ .cos θ+ 25 sin² θ + 9 cos² θ – 30 sin θ . cos θ = 25 + x²
⇒ 9 sin²θ + 9 cos²θ + 25 cos2 θ + 25 sin²θ = 25 + x²
⇒ 9 (sin²θ + cos²θ) + 25 (cos²θ + sin²θ ) = 25 + x²
⇒ 9 + 25 = 25 + x²
⇒ x² = 9
⇒ x = ± 3