If Sec θ = 13/12 for an acute angle θ in a right triangle, what is the exact value of Sin θ?

Introduction / Context:
This question tests the ability to convert between different trigonometric ratios by using the right triangle definition and the Pythagorean identity. Knowing Sec θ allows us to find Cos θ, then Sin θ, which is crucial in many problems involving right triangles and trigonometric simplification.

Given Data / Assumptions:

• Sec θ = 13/12.
• Angle θ is acute, so all basic trig ratios are positive.
• Secant is the reciprocal of cosine.
• We need to find Sin θ.

Concept / Approach:
Since Sec θ = 1 / Cos θ, we can find Cos θ as 12/13. In a right triangle, Cos θ = adjacent / hypotenuse. With these values, we can use the Pythagorean theorem to find the opposite side. Then Sin θ = opposite / hypotenuse. This geometric approach is direct and avoids unnecessary symbolic manipulation.

Step-by-Step Solution:
Given Sec θ = 13/12, so Cos θ = 1 / Sec θ = 12/13. Interpret Cos θ = adjacent / hypotenuse = 12/13. Let adjacent side be 12 units and hypotenuse be 13 units. Use the Pythagorean theorem: hypotenuse^2 = adjacent^2 + opposite^2. Compute 13^2 = 12^2 + opposite^2, so 169 = 144 + opposite^2. Thus opposite^2 = 169 − 144 = 25. So opposite side = 5 units (positive because θ is acute). Now Sin θ = opposite / hypotenuse = 5 / 13.

Verification / Alternative check:
Check the fundamental identity Sin^2 θ + Cos^2 θ = 1. With Sin θ = 5/13 and Cos θ = 12/13, we have (5/13)^2 + (12/13)^2 = 25/169 + 144/169 = 169/169 = 1. This confirms that the pair of ratios is consistent with a valid right triangle and that our computation is correct.

Why Other Options Are Wrong:
12/13 is Cos θ, not Sin θ. The ratio 5/12 would correspond to opposite over adjacent, more like the tangent value in this triangle. The values 12/5 and 13/5 are greater than 1, which is impossible for the sine of an acute angle. Thus only 5/13 matches the correct definition for Sin θ in this context.

Common Pitfalls:
Some learners mistakenly treat Sec θ itself as opposite over hypotenuse or confuse it with Cos θ. Others directly guess numerical values without constructing the triangle or checking the identity. Drawing a quick sketch and labelling sides based on the ratio significantly improves accuracy in such problems.

Final Answer:
The exact value of Sin θ is 5/13.