Question

In \( \triangle ABC \), if \( \sin B = \sin C \) and \( 3 \cos B = 2 \cos C \), then \( \triangle ABC \) is:

• A. a right-angled triangle
• B. an isosceles triangle
• C. an equilateral triangle
• D. a scalene triangle

Hint:

In triangles, if two angles are equal, the triangle is isosceles. If the third angle is different, it is scalene.

Answer 1

The Correct Option is D

We are given that: \[ \sin B = \sin C \quad \text{and} \quad 3 \cos B = 2 \cos C \] From \( \sin B = \sin C \), we conclude that: \[ B = C \quad \text{(since \( B \) and \( C \) are angles of a triangle, and the sine function is positive in the first quadrant).} \] From \( 3 \cos B = 2 \cos C \), substituting \( B = C \), we get: \[ 3 \cos B = 2 \cos B \] This simplifies to: \[ \cos B = 0 \] Thus, \( B = 90^\circ \), making \( \triangle ABC \) a right-angled triangle. Since \( B = C \), \( \triangle ABC \) is isosceles and scalene. % Final Answer \[ \boxed{\text{Scalene Triangle}} \]