<p>93. If in a △ABC, c = 150, b = 50√3 and B = 30° then C has the measure</p>
Step-by-Step Solution
Key Concept: Use the sine rule (a/sin A = b/sin B = c/sin C) to find angle C directly from the given sides and angle. The key is recognizing that we can find sin C without finding side a first.
<p><strong>Step 1:</strong> Apply the sine rule: b/sin B = c/sin C</p><p><strong>Step 2:</strong> Substitute values: (50√3)/sin 30° = 150/sin C</p><p><strong>Step 3:</strong> Since sin 30° = 1/2: (50√3)/(1/2) = 150/sin C</p><p>100√3 = 150/sin C</p><p><strong>Step 4:</strong> Solve for sin C: sin C = 150/(100√3) = 3/(2√3) = √3/2</p><p><strong>Step 5:</strong> From sin C = √3/2, we get C = 60° or C = 120°</p><p><strong>Step 6:</strong> Check validity: Since c = 150 > b = 50√3 ≈ 86.6, the larger side c must be opposite the larger angle. With B = 30°, if C = 60° then A = 90° (valid). If C = 120° then A = 30°, but then c should be largest (it is), yet B = A = 30° with sides b ≠ a (contradiction).</p><p><strong>Step 7:</strong> The ambiguous case resolves: C = 60° is the answer since the triangle with B = 30°, C = 60°, A = 90° satisfies all constraints.</p><p>∴ Answer: C = 60°</p>
Correct Answer: A