If θ be positive acute angle and 5 cosθ + 12

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If θ be positive acute angle and 5 cosθ + 12 sinθ = 13, then the value of cosθ is

1. 12
  13
2. 5
  13
3. 5
  12
4. 1
  5

Correct Option: B

5 cosθ + 12 sinθ = 13
⇒ 5 cosθ – 13 = 12 sinθ
On squaring both sides,
25 cos²θ + 169 – 130 cosθ
= 144 (1 – cos²θ)
⇒ cos²θ – 130 cosθ + 169
= 144 – 144 cos²θ
⇒ 144 cos²θ + 25 cos²θ – 130 cosθ + 169 – 144 = 0
⇒ 169 cos²θ – 130 cosθ + 25 = 0
⇒ (13 cosθ – 5)² = 0
⇒ 13 cosθ – 5 = 0
⇒ 13 cosθ = 5

⇒ cosθ =	5
	13

OR
5 cosθ + 12 sinθ = 13

⇒	5	cosθ +	12	sinθ = 1
	13		13

∵ cos²θ + sin²θ = 1

∴ cosθ =	5
	13