Difficulty: Easy
Correct Answer: 17/8
Explanation:
Introduction / Context:
This question links cosine and sine through the Pythagorean identity and then uses the definition of cosecant. It checks understanding of basic trigonometric relationships in right triangles when one ratio is known.
Given Data / Assumptions:
Concept / Approach:
Recall the identity sin² θ + cos² θ = 1. From cos θ we compute sin θ. Then cosec θ is simply 1 divided by sin θ. Because θ is acute, both sine and cosine are positive, so we take the positive square root when finding sin θ.
Step-by-Step Solution:
Given cos θ = 15/17
Use identity sin² θ + cos² θ = 1
So sin² θ = 1 − cos² θ
cos² θ = (15/17)² = 225/289
Hence sin² θ = 1 − 225/289 = (289 − 225)/289 = 64/289
Since θ is acute, sin θ is positive, so sin θ = 8/17
Then cosec θ = 1 / sin θ = 1 / (8/17) = 17/8
Verification / Alternative check:
We can imagine a right triangle with adjacent side 15 units and hypotenuse 17 units. By Pythagoras, the opposite side is √(17² − 15²) = √(289 − 225) = √64 = 8 units. So sin θ = opposite/hypotenuse = 8/17 and cosec θ is hypotenuse/opposite = 17/8. This geometric view confirms the algebraic result.
Why Other Options Are Wrong:
8/17 is the value of sin θ, not cosec θ. 8/15 and 17/15 correspond to other improper ratios not tied to sin or cos in this setting. 15/17 is again cos θ and not its reciprocal counterpart. Only 17/8 matches the definition of cosec θ for the given triangle.
Common Pitfalls:
Some learners incorrectly think that if cos θ is 15/17, then sin θ must be 17/15 or confuse which side is opposite or adjacent. Others forget that trigonometric ratios for acute angles are positive, mistakenly choosing negative roots when square roots appear. Keeping track of geometric meaning avoids these issues.
Final Answer:
Thus, the value of cosec θ is 17/8.
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