If secθ = 29/20 where 0

Question

If secθ = 29/20 where 0 < θ < 90°, then what is the value of 3cosecθ + 3cotθ?

Accepted Answer

Given secθ = 29/20 and 0 < θ < 90°, we need to find 3cosecθ + 3cotθ.

Step 1: Find cosθ.
secθ = 1/cosθ ⇒ cosθ = 1/secθ = 20/29.

Step 2: Find sinθ.
Since 0 < θ < 90°, sinθ > 0.
Using sin²θ + cos²θ = 1,
sin²θ = 1 - cos²θ = 1 - (20/29)² = 1 - 400/841 = (841 - 400)/841 = 441/841.
So, sinθ = √(441/841) = 21/29.

Step 3: Find cosecθ and cotθ.
cosecθ = 1/sinθ = 29/21.
cotθ = cosθ/sinθ = (20/29) / (21/29) = 20/21.

Step 4: Calculate 3cosecθ + 3cotθ.
= 3 * (29/21) + 3 * (20/21) = (3*29 + 3*20)/21 = (87 + 60)/21 = 147/21 = 7.

Therefore, the value is 7.

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