Trigonometry & Inverse Trigonometry
Triangle Properties
Grade 11
Question:
<p>Angles <i>A</i>, <i>B</i> and <i>C</i> of a △<i>ABC</i> are in AP. If \(\frac{b}{c} = \frac{\sqrt{3}}{2}\), then ∠<i>A</i> is equal to</p>
<p>(a) \(\frac{7\pi}{6}\)</p>
<p>(b) \(\frac{\pi}{4}\)</p>
<p>(c) \(\frac{5\pi}{6}\)</p>
<p>(d) \(\frac{\pi}{12}\)</p>
Step-by-Step Solution
Key Concept: Use the arithmetic progression condition and sine rule to determine the angles.
<p>Since <i>A</i>, <i>B</i>, <i>C</i> are in AP: $2B = A + C$</p><p>Also, $A + B + C = \pi$, so $B = \frac{\pi}{3}$</p><p>Using sine rule: $\frac{b}{c} = \frac{\sin B}{\sin C} = \frac{\sqrt{3}}{2}$</p><p>This gives $\sin C = \frac{2\sin B}{\sqrt{3}} = \frac{2 \cdot \frac{\sqrt{3}}{2}}{\sqrt{3}} = 1$, so $C = \frac{\pi}{2}$</p><p>Therefore, $A = \pi - B - C = \pi - \frac{\pi}{3} - \frac{\pi}{2} = \frac{\pi}{6}$</p>
Correct Answer: B