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: Understanding the Concept
In any triangle \(A+B+C=\pi\), so \(\sin A=\sin(B+C)\).

Step 2: Key Formula or Approach
The given condition gives \(2\sin B\cos C=\sin A\).

Step 3: Detailed Explanation
\[ 2\sin B\cos C=\sin B\cos C+\cos B\sin C \]
\[ \sin B\cos C-\cos B\sin C=0 \Rightarrow \sin(B-C)=0 \]
Since \(B-C\) lies between \(-\pi\) and \(\pi\), \(B=C\). Equal angles mean equal opposite sides, so \(b=c\).

Final Answer:
The triangle has \(b=c\), option (C). \[ \boxed{b=c\ \text{(C)}} \]