Difficulty: Easy
Correct Answer: 29/21
Explanation:
Introduction / Context:
This trigonometry question tests your understanding of basic definitions of sine, cosine, and secant in a right triangle. Given sin θ, you are asked to find sec θ, which requires you to relate the opposite, adjacent, and hypotenuse sides and use the Pythagorean identity. This kind of conversion between trigonometric ratios appears frequently in aptitude exams.
Given Data / Assumptions:
Concept / Approach:
By definition, sin θ = opposite / hypotenuse. Once we interpret 20/29 as side lengths, we can use the Pythagorean theorem to find the remaining side. From there, cos θ = adjacent / hypotenuse and sec θ = 1 / cos θ. Because θ is acute, we choose the positive square root when we compute the missing side length.
Step-by-Step Solution:
Verification / Alternative check:
As a quick check, confirm that the sides 20, 21, and 29 satisfy the Pythagorean theorem: 20^2 + 21^2 = 400 + 441 = 841, and 29^2 = 841. The identity holds, which means the triangle is consistent. Using these sides, sin θ = 20/29 and sec θ = 29/21 are correctly related by trigonometric definitions.
Why Other Options Are Wrong:
The value 29/20 would imply cos θ = 20/29, which is incorrect based on the side lengths. The values 21/20 and 21/29 correspond to incorrect ratios of sides and do not match 1 / cos θ. The fraction 20/29 is simply sin θ, not sec θ. Only 29/21 is equal to the reciprocal of cos θ for this triangle.
Common Pitfalls:
Students sometimes confuse sine and cosine and mistakenly interchange opposite and adjacent. Another common mistake is forgetting that sec θ is the reciprocal of cos θ, not of sin θ. Using a clear diagram and the Pythagorean theorem helps to avoid mixing up the sides and the corresponding ratios.
Final Answer:
The exact value of sec θ for sin θ = 20/29 is 29/21.
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