if cos theta + sec theta is equal to 10 /3 find the value of sin theta + cos 8 theta
Answers
Given : cosθ + sec θ= 10 / 3
θ is acute angle
To Find : sinθ + cosecθ
Solution:
cosθ + sec θ= 10 / 3
=> cosθ + 1/cos θ= 10 / 3
=> 3cos²θ + 3 = 10cos θ
=> 3cos²θ - 10cos θ + 3 = 0
=> 3cos²θ - 9cos θ - cos θ + 3 = 0
=> 3cos θ(cos θ - 3) - 1( cos θ - 3) = 0
=> (3cos θ - 1)( cos θ - 3) = 9
=> cos θ = 1/3 , Cosθ = 3
Cosθ lies between - 1 and 1
hence cos θ = 1/3
cos²θ + sin²θ = 1
=> (1/3)² + sin²θ = 1
=> sin²θ = 8/9
=> sinθ = 2√2/3 as θ is acute hence sinθ is +ve
sinθ + cosecθ = sinθ + 1/sinθ
= 2√2/3 + 3/2√2
= (8 + 9)/6√2
= 17/6√2
≈ 2.003
sinθ + cosecθ = 17/6√2 ≈ 2.003
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