Trigonometry & Inverse Trigonometry
Sine Rule
Grade 11
Question:
<p>In triangle ABC, given a/sin A = 2√2/sin 30° = 4/sin C. Find angles C and A.</p>
<p>(A) C = 30°, A = 120°</p>
<p>(B) C = 45° or 135°; A = 105° or 15°</p>
<p>(C) C = 60°, A = 90°</p>
<p>(D) C = 90°, A = 60°</p>
Step-by-Step Solution
Key Concept: Apply the sine rule carefully and recognize that sine has two possible angles in the range [0°, 180°].
<p><strong>Step 1:</strong> Using the sine rule:<br>a/sin A = 2√2/sin 30° = 4/sin C</p><p><strong>Step 2:</strong> From 2√2/sin 30° = 4/sin C:<br>2√2/(1/2) = 4/sin C<br>4√2 = 4/sin C<br>sin C = 1/√2 = √2/2</p><p><strong>Step 3:</strong> This gives C = 45° or C = 135°</p><p><strong>Step 4:</strong> Using A + B + C = 180° with B = 30°:<br>When C = 45°: A = 180° - 30° - 45° = 105°<br>When C = 135°: A = 180° - 30° - 135° = 15°</p><p>∴ Answer is (B): C = 45° or 135°; A = 105° or 15°</p>
Correct Answer: B