Question

In \(△ABC\), with usual notations, if \(cosA = \frac{1}{2},a = 3,\angle B = \frac{π^c}{6}\), then the values of \(b\) and \(c\) are.....

• A. \(b = \sqrt{3},c = \sqrt{3}\)
• B. \(b = 2\sqrt{3},c = \sqrt{3}\)
• C. \(b = \sqrt{3},c = 2\sqrt{3}\)
• D. \(b = \sqrt{3},c = \frac{1}{\sqrt{3}}\)

Hint:

Find angle $A$, then $C$, then use the sine rule.

Answer 1

The Correct Option is C

Step 1: Find the angles
\(\cos A=\frac12\Rightarrow A=60^{\circ}\). With \(B=30^{\circ}\), \(C=180^{\circ}-90^{\circ}=90^{\circ}\).

Step 2: Use the sine rule
\[ \frac{a}{\sin A}=\frac{3}{\sqrt3/2}=2\sqrt3 \]

Step 3: Find \(b\) and \(c\)
\(b=2\sqrt3\sin30^{\circ}=\sqrt3\).
\(c=2\sqrt3\sin90^{\circ}=2\sqrt3\).
This is option (C).

Final Answer:
\(b=\sqrt3\) and \(c=2\sqrt3\), option (C). \[ \boxed{\text{(C)}} \]