If 2cosec²θ + 3cot²θ = 17, then find the value of θ when 0° ≤ θ ≤ 90°.

Question

Accepted Answer

Given: 2cosec²θ + 3cot²θ = 17

Step 1: Use the identity cot²θ = cosec²θ - 1.

Rewrite the equation:
2cosec²θ + 3(cosec²θ - 1) = 17

Step 2: Expand and simplify:
2cosec²θ + 3cosec²θ - 3 = 17
(2 + 3)cosec²θ - 3 = 17
5cosec²θ = 20
cosec²θ = 20 / 5 = 4

Step 3: Find cosecθ:
cosecθ = ±2
Since 0° ≤ θ ≤ 90°, cosecθ is positive, so cosecθ = 2

Step 4: Find sinθ:
sinθ = 1 / cosecθ = 1 / 2 = 0.5

Step 5: Find θ:
θ = sin⁻¹(0.5) = 30°

Therefore, the value of θ is 30°.

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