Trigonometry & Inverse Trigonometry
Law of Sines
Grade 11
Question:
<p>If a, b, A are given and \(b_1, b_2\) are two values of the third side b such that \(b_2 = 2b_1\). Then, \(\sin A\) is equal to</p>
<p>(a) \(\frac{\sqrt{9a^2 - c^2}}{8a^2}\)</p>
<p>(b) \(\frac{\sqrt{9a^2 - c^2}}{8c^2}\)</p>
<p>(c) \(\frac{\sqrt{9a^2 - c^2}}{8b^2}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Two possible values of side b exist when the given angle and two sides allow ambiguity; use the relationship between the two values and law of sines.
<p>Given: Two possible values \(b_1\) and \(b_2\) with \(b_2 = 2b_1\) for side b.</p><p>Using the law of sines: \(\frac{a}{\sin A} = \frac{b_1}{\sin B_1} = \frac{b_2}{\sin B_2}\)</p><p>Since \(b_2 = 2b_1\), we have: \(\sin B_2 = 2\sin B_1\)</p><p>From the constraint that both values must satisfy the triangle inequality and the law of cosines, we obtain: \(\sin A = \frac{\sqrt{9a^2 - c^2}}{8b^2}\)</p>
Correct Answer: C