If sec θ = 5/3 for an acute angle θ in a right triangle, then what is the value of cosec θ?

Introduction / Context:
This trigonometry question links secant and cosecant through sine and cosine. Given sec θ, you can determine cos θ, then use the Pythagorean identity to find sin θ, and finally obtain cosec θ as the reciprocal of sin θ. It is a standard exercise in relating different trigonometric ratios for an acute angle.

Given Data / Assumptions:

• sec θ = 5/3.
• θ is an acute angle (0° < θ < 90°).
• sec θ = 1 / cos θ and cosec θ = 1 / sin θ.
• We use sin² θ + cos² θ = 1.

Concept / Approach:
From sec θ = 5/3 we immediately get cos θ = 3/5. Then we use the Pythagorean identity sin² θ + cos² θ = 1 to compute sin θ. Because θ is acute, sin θ will be positive. Once sin θ is known, cosec θ is simply the reciprocal of sin θ. We can also interpret this geometrically in a right triangle for additional intuition.

Step-by-Step Solution:
Given sec θ = 5/3. Recall sec θ = 1 / cos θ, so cos θ = 3/5. Use identity sin² θ + cos² θ = 1. So sin² θ = 1 − cos² θ = 1 − (3/5)². (3/5)² = 9/25, so sin² θ = 1 − 9/25 = 16/25. Since θ is acute, sin θ is positive, so sin θ = 4/5. Now cosec θ = 1 / sin θ = 1 / (4/5) = 5/4. As a decimal, 5/4 = 1.25.

Verification / Alternative check:
Visualize a right triangle with hypotenuse 5 units and adjacent side 3 units. Then cos θ = 3/5, matching the given sec θ = 5/3. The opposite side by Pythagoras is √(5² − 3²) = √(25 − 9) = √16 = 4, so sin θ = 4/5. The reciprocal of 4/5 is 5/4, confirming the value of cosec θ. As a decimal, this is 1.25, which aligns with the correct option given.

Why Other Options Are Wrong:
0.8 is 4/5, the sine itself rather than its reciprocal. 4/3 and 3/4 correspond to other ratios not linked to the computed sine in this context. The decimal 0.8 and the fraction 3/4 are both less than 1, while cosec θ must be greater than or equal to 1 for acute angles. Only 1.25 (which equals 5/4) matches the derived value of cosec θ.

Common Pitfalls:
Learners often confuse sec θ with cos θ or mistakenly take sec θ as cos θ directly. Another mistake is to forget that for acute angles, we always take the positive square root when evaluating sin² θ. A few may incorrectly invert cos θ instead of sin θ for cosec θ. Following the identity and definitions step by step avoids these pitfalls.

Final Answer:
Thus, the value of cosec θ is 1.25 (that is, 5/4).