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Question

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BlackTom AI · Accepted Answer

The problem provides two angles A = 75° and B = 40°, and side b = 20 m opposite angle B. We need to find side a opposite angle A.

First, compute the remaining angle C using the angle sum of a triangle: C = 180° − (A + B) = 180° − (75° + 40°) = 65°.

Next, apply the Law of Sines, which states a / sin(A) = b / sin(B). Solving for a gives a = b * sin(A) / sin(B).

Calculate the sine values (approximate): sin(75°) ≈ 0.9659 and sin(40°) ≈ 0.6428. Thus a ≈ 20 * (0.9659 / 0.6428) ≈ 20 * 1.503 ≈ 30.06 m.

Now examine the answer options:
- Option a: 0.03 m. This is far too small compared to the computed value and would imply an almost degenerate triangle; it arises if one mistakenly divides or uses incorrect sine values. Eliminated.
- Option b: 13.3 m. This is substantially less than the computed 30 m. It could result from misplacing the ratio or using a wrong angle, such as using sin(B)/sin(A) instead of sin(A)/sin(B). Eliminated.
- Option c: 32.2 m. This is in the right ballpark but slightly larger than the calculated 30.1 m; such a discrepancy could come from rounding sin values too aggressively or carrying extra digits. Note that using more precise sines brings you closer to the computed value.
- Option d: 30.1 m. This matches our calculated value to the given precision (about 30.06 m), and aligns with the Law of Sines calculation.

Overall, the correct reasoning uses the Law of Sines with b opposite B and a opposite A, showing why the value close to 30.1 m satisfies the relationship a / sin(75°) = b / sin(40°). The other options stem from common pitfalls such as misplacing the sine ratio or misinterpreting the given angle-side pair, leading to values that are either drastically too small, too large, or only loosely approximate despite using basic trigonometric principles.

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