Difficulty: Medium
Correct Answer: 15/8
Explanation:
Introduction / Context:
This trigonometry problem uses the relationships between the trigonometric ratios of complementary angles in a right triangle. You are given cosec X and asked to find cot Z, where X and Z are the two acute angles in the triangle. Understanding the side labeling in a right triangle and how ratios for different angles relate to the same sides is the core skill tested here.
Given Data / Assumptions:
Concept / Approach:
The definition of cosec X is hypotenuse / opposite to angle X. From cosec X = 17/15, we can identify the lengths of the hypotenuse and the side opposite to X. Then we use the Pythagorean theorem to find the remaining side. Since Z is the other acute angle, the roles of opposite and adjacent sides swap when we consider angle Z. Finally, we use cot Z = adjacent / opposite for angle Z.
Step-by-Step Solution:
Verification / Alternative check:
You can confirm the relationships by checking that angles X and Z are complementary. For angle X, sin X = opposite / hypotenuse = 15/17, so cosec X = 17/15, which matches the given value. For angle Z, tan Z would be opposite / adjacent = 8/15, so cot Z = 15/8. This consistency confirms that our side assignments and calculations are correct.
Why Other Options Are Wrong:
The value 17/15 is cosec X, not cot Z. The values 8/17 and 17/8 correspond to other trigonometric ratios such as sin Z or cot X. The fraction 8/15 is tan Z, the reciprocal of cot Z. Only 15/8 matches the definition of cotangent for angle Z in this triangle.
Common Pitfalls:
Learners can become confused about which side is opposite or adjacent when changing from one angle to another in the same triangle. Some also mistakenly treat cosec as opposite / hypotenuse instead of its reciprocal. Carefully drawing a labelled triangle and tracking which side corresponds to which angle helps to avoid such confusion.
Final Answer:
The required value of cot Z is 15/8.
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