Given a trigonometric ratio for an acute angle, you can build a right triangle and find other ratios using Pythagoras.\nIf cosec θ = 17/8 for an acute angle θ, what is the value of cos θ?

Introduction / Context:
This question tests converting from cosec θ to cos θ using a right-triangle model. Since θ is acute, all ratios are positive. From cosec θ, we obtain sin θ, interpret it as opposite/hypotenuse, use Pythagoras to find the adjacent side, and then compute cos θ = adjacent/hypotenuse.

Given Data / Assumptions:

• cosec θ = 17/8
• θ is acute, so sin θ > 0 and cos θ > 0
• cosec θ = 1/sin θ

Concept / Approach:
Convert cosec to sin: sin θ = 8/17. Treat this as a right triangle where opposite = 8 and hypotenuse = 17. Then adjacent = sqrt(17^2 - 8^2) = 15, giving cos θ = 15/17.

Step-by-Step Solution:
cosec θ = 17/8 => sin θ = 1/(17/8) = 8/17 Let opposite = 8 and hypotenuse = 17 Adjacent^2 = hypotenuse^2 - opposite^2 = 17^2 - 8^2 = 289 - 64 = 225 Adjacent = sqrt(225) = 15 cos θ = adjacent/hypotenuse = 15/17

Verification / Alternative check:
Since sin θ = 8/17 and cos θ = 15/17, check sin^2 + cos^2 = (64 + 225)/289 = 289/289 = 1, consistent with a right triangle.

Why Other Options Are Wrong:
8/17 is sin θ, not cos θ.
17/15 is sec θ (reciprocal of cos).
8/15 is tan θ for the 8-15-17 triangle.
15/8 is cosec-related or an inverted ratio and cannot be cos because it is greater than 1.

Common Pitfalls:
Inverting incorrectly (mixing cosec and sec), or assuming cos θ = 1/cosec θ (which is wrong). Also, some forget to compute the adjacent side using Pythagoras and guess using the given numbers.

Final Answer:
cos θ = 15/17