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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