In a right triangle with acute angle θ, if Cos θ = 35/37, then what is the exact value of Cosec θ?

Introduction / Context:
This trigonometry question examines the relationship between cosine, sine, and cosecant in a right triangle. Given a value for Cos θ, we can reconstruct the triangle sides and then find Sin θ and Cosec θ. This type of problem helps build geometric intuition for trigonometric ratios beyond simple memorisation of values.

Given Data / Assumptions:

• Cos θ = 35/37 for an acute angle θ.
• Cos θ is defined as adjacent side divided by hypotenuse in a right triangle.
• We need Cosec θ, which is the reciprocal of Sin θ.
• Angle θ is acute, so all basic trigonometric ratios are positive.

Concept / Approach:
If Cos θ = adjacent / hypotenuse = 35 / 37, we can model a right triangle where the adjacent side is 35 units and the hypotenuse is 37 units. Then we use the Pythagorean theorem to find the length of the opposite side. After that we compute Sin θ = opposite / hypotenuse, and finally find Cosec θ = 1 / Sin θ. This geometric method is straightforward and avoids unnecessary algebraic manipulation of trigonometric identities.

Step-by-Step Solution:
Interpret Cos θ = 35/37 as adjacent = 35 and hypotenuse = 37. Use the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2. So 37^2 = 35^2 + opposite^2. Compute 37^2 = 1369 and 35^2 = 1225. Then opposite^2 = 1369 − 1225 = 144, so opposite = 12 (positive for acute angle). Therefore Sin θ = opposite / hypotenuse = 12 / 37. Cosec θ is the reciprocal of Sin θ, so Cosec θ = 37 / 12.

Verification / Alternative check:
We can verify consistency by computing Cos^2 θ + Sin^2 θ. With Cos θ = 35/37 and Sin θ = 12/37, we get (35/37)^2 + (12/37)^2 = (1225 + 144)/1369 = 1369/1369 = 1, which satisfies the fundamental identity Sin^2 θ + Cos^2 θ = 1. This confirms that the triangle sides and the resulting cosecant value are correct.

Why Other Options Are Wrong:
33/12 and 35/12 do not match the reciprocal of 12/37. The value 12/35 corresponds more closely to a tangent or cotangent type ratio and not to cosecant in this setup. The fraction 37/35 mixes the hypotenuse and adjacent sides wrongly. Only 37/12 correctly represents the reciprocal of Sin θ for this triangle.

Common Pitfalls:
Sometimes students swap the roles of opposite and adjacent, especially if they do not sketch the triangle first. Another error is arithmetic, such as computing 37^2 or 35^2 incorrectly or miscalculating the difference. Drawing a quick diagram and verifying Sin^2 θ + Cos^2 θ = 1 are simple ways to avoid these mistakes.

Final Answer:
The exact value of Cosec θ is 37/12.