For an acute angle θ, if sin θ = 20/29, model θ using a right triangle, apply the Pythagoras theorem to find the remaining side, and then determine the exact value of cos θ.

Introduction / Context:
This trigonometry question asks you to find cos θ when sin θ is given for an acute angle. Instead of using inverse trigonometric functions, you can treat the sine value as a ratio of sides in a right triangle and then use the Pythagoras theorem to find the adjacent side. This approach is straightforward and commonly used in competitive exams for speed and accuracy.

Given Data / Assumptions:

• sin θ = 20/29.
• θ is an acute angle, so 0 degrees < θ < 90 degrees.
• sin θ = opposite side / hypotenuse.
• cos θ = adjacent side / hypotenuse.
• Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2.

Concept / Approach:
Interpret sin θ = 20/29 by constructing a right triangle where the side opposite θ is 20 units and the hypotenuse is 29 units. Then use the Pythagoras theorem to find the adjacent side. Once all sides are known, cos θ is simply the ratio of adjacent side to hypotenuse. Because θ is acute, all lengths are positive and we select the positive root when applying square roots.

Step-by-Step Solution:
Let opposite side to θ be 20 and hypotenuse be 29. Apply Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2. Substitute: 29^2 = 20^2 + adjacent^2. Compute: 29^2 = 841 and 20^2 = 400. Thus adjacent^2 = 841 - 400 = 441. Take the square root: adjacent = 21, since 21^2 = 441. Now cos θ = adjacent / hypotenuse = 21/29.
Verification / Alternative check:
You can verify that sin^2 θ + cos^2 θ = 1, as required by the basic identity. Compute sin^2 θ = (20/29)^2 = 400/841 and cos^2 θ = (21/29)^2 = 441/841. Their sum is (400 + 441)/841 = 841/841 = 1, confirming that cos θ = 21/29 is consistent with the given sine value and the Pythagoras identity.

Why Other Options Are Wrong:
Option b (29/21) is simply the reciprocal of the correct answer and would correspond to sec θ, not cos θ. Option c (21/20) is a ratio of adjacent to opposite, not adjacent to hypotenuse. Option d (20/29) is sin θ itself, not cos θ. Option e (29/20) is the reciprocal of sin θ, that is cosec θ, and therefore not the required cosine value.

Common Pitfalls:
A frequent mistake is to assign the hypotenuse and opposite sides incorrectly, for example treating 20 as hypotenuse and 29 as opposite, which violates the basic property that the hypotenuse is the longest side. Another pitfall is arithmetic error when squaring 29 or subtracting 400 to find the adjacent squared value. Carefully computing 29^2 = 841 and subtracting 400 to get 441 ensures you obtain the correct adjacent length and thus the correct cosine.

Final Answer:
The exact value of cos θ when sin θ = 20/29 for an acute angle is 21/29.