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If A is an acute angle and sin A = x, which of the following expressions is exactly equal to x?

Difficulty: Easy

Correct Answer: sqrt(1 - cos^2 A)

Explanation:


Introduction / Context:
This question tests the Pythagorean identity sin^2 A + cos^2 A = 1 and how to express sin A in terms of cos A. The condition that A is an acute angle is important because it ensures sin A is positive, so sin A equals the positive square root of sin^2 A. Without the acute-angle condition, we would need a plus/minus sign.


Given Data / Assumptions:

  • A is an acute angle (0° < A < 90°).
  • sin A = x.
  • Pythagorean identity: sin^2 A + cos^2 A = 1.


Concept / Approach:
From sin^2 A + cos^2 A = 1, isolate sin^2 A. Then take the square root. Because A is acute, sin A is positive, so we use only the positive root.


Step-by-Step Solution:

Step 1: Start with sin^2 A + cos^2 A = 1. Step 2: Rearrange to isolate sin^2 A: sin^2 A = 1 - cos^2 A. Step 3: Take square root on both sides: sin A = sqrt(1 - cos^2 A) or sin A = -sqrt(1 - cos^2 A). Step 4: Since A is acute, sin A > 0, so sin A = sqrt(1 - cos^2 A). Step 5: Given sin A = x, we get x = sqrt(1 - cos^2 A).


Verification / Alternative check:
Take A = 30°: sin 30° = 1/2. cos 30° = sqrt(3)/2. Then sqrt(1 - cos^2 30°) = sqrt(1 - 3/4) = sqrt(1/4) = 1/2, which matches sin 30°.


Why Other Options Are Wrong:

sqrt(1 - sin^2 A): equals |cos A|, not sin A. sqrt(1 + cos^2 A) and sqrt(1 + sin^2 A): these are always greater than or equal to 1, so they cannot equal sin A for an acute angle. sqrt(1 - cos A): unrelated identity; does not match sin A generally.


Common Pitfalls:
Forgetting the plus/minus when taking square roots, and ignoring the acute-angle condition that removes the negative possibility.


Final Answer:
sqrt(1 - cos^2 A)

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