If cos θ = 5/13 for an acute angle θ in a right triangle, then using the Pythagorean identity, what is the exact value of cosec θ?

Introduction / Context:
This question links cosine and cosecant using the fundamental Pythagorean identity sin² θ + cos² θ = 1. Given cos θ for an acute angle, you can determine sin θ and then compute its reciprocal to obtain cosec θ. This is a standard exercise in basic trigonometry, especially in problems involving right triangles and exact ratios.

Given Data / Assumptions:

• cos θ = 5/13.
• θ is an acute angle (0° < θ < 90°).
• sin² θ + cos² θ = 1.
• cosec θ = 1 / sin θ.

Concept / Approach:
From cos θ we can imagine a right triangle where the adjacent side is 5 units and the hypotenuse is 13 units. Using Pythagoras, the opposite side can be found as √(13² − 5²). This gives sin θ as opposite/hypotenuse and therefore cosec θ as hypotenuse/opposite. Alternatively, we can find sin² θ directly from the identity sin² θ = 1 − cos² θ and then take the positive square root because θ is acute.

Step-by-Step Solution:
Given cos θ = 5/13. Compute cos² θ = (5/13)² = 25/169. Use identity sin² θ + cos² θ = 1. So sin² θ = 1 − cos² θ = 1 − 25/169. Write 1 as 169/169: sin² θ = 169/169 − 25/169 = 144/169. Since θ is acute, sin θ = √(144/169) = 12/13. Now cosec θ = 1 / sin θ = 1 / (12/13) = 13/12.

Verification / Alternative check:
Visualize the triangle: adjacent side = 5, hypotenuse = 13. Then the opposite side is √(13² − 5²) = √(169 − 25) = √144 = 12. Thus sin θ = opposite/hypotenuse = 12/13, and cosec θ = hypotenuse/opposite = 13/12. This geometric interpretation perfectly matches the algebraic calculation and confirms the answer.

Why Other Options Are Wrong:
5/12 and 12/13 represent cos/sin or sin/cos combinations but not cosec θ here. 12/5 is the reciprocal of 5/12 and does not correspond to any primary ratio for θ given cos θ = 5/13. The value 13/5 is much larger and would not match the reciprocal of sin θ. Only 13/12 is consistent with the derived value of cosec θ for this triangle.

Common Pitfalls:
Some learners may attempt to invert cos θ directly to get cosec θ, which is incorrect because cosec θ is the reciprocal of sin θ, not cos θ. Others might mistakenly choose the negative square root when solving for sin θ, forgetting that sin θ is positive in the first quadrant. Carefully applying the Pythagorean identity and the definitions avoids these errors.

Final Answer:
Hence, the value of cosec θ is 13/12.