Question

With usual notations, in \(△ABC\), if \(cosC = \frac{sinA}{2sinB}\), then which of the following is true ?

• A. \(a = c\)
• B. \(a = b\)
• C. \(b = c\)
• D. \(a^2 = b^2+c^2\)

Hint:

Write \(\sin A=\sin(B+C)\) and simplify.

Answer 1

The Correct Option is C

Step 1: Use the sine and cosine rules
$\cos C=\dfrac{a^2+b^2-c^2}{2ab}$ and $\dfrac{\sin A}{\sin B}=\dfrac{a}{b}$.

Step 2: Substitute
The condition becomes $\dfrac{a^2+b^2-c^2}{2ab}=\dfrac{a}{2b}$, so $a^2+b^2-c^2=a^2$, giving $b^2=c^2$ and $b=c$. Option (C).

Final Answer:
We get $b=c$, option (C). \[ \boxed{b=c} \]