Question

In \( \triangle ABC \), if \[ a \cos^2 \frac{C}{2} + \cos^2 \frac{A}{2} = \frac{3b}{2}, \] then \( a + c : b \) is:

• A. \( 1 : 1 \)
• B. \( 3 : 2 \)
• C. \( 2 : 1 \)
• D. \( 4 : 3 \)

Hint:

In problems involving trigonometric identities in triangles, use known formulas and simplify step by step to obtain the required ratio.

Answer 1

The Correct Option is C

We are given: \[ a \cos^2 \frac{C}{2} + \cos^2 \frac{A}{2} = \frac{3b}{2} \] Step 1: Apply the cosine rule and trigonometric identities in terms of sides and angles of the triangle. We can use the identity for \( \cos \frac{C}{2} \) and \( \cos \frac{A}{2} \) and express the relationship between the sides \( a \), \( b \), and \( c \). Step 2: Simplify the equation and express the ratio \( a + c : b \). Using basic algebraic manipulation and trigonometric relationships, we find that the ratio is: \[ a + c : b = 2 : 1 \] % Final Answer \[ \boxed{2 : 1} \]