Question

In a triangle ABC, if $a, b, c$ are in arithmetic progression and the angle $A$ is twice the angle $C$, then $\cos A : \cos B : \cos C =$

• A. $2 : 3 : 4$
• B. $3 : 4 : 8$
• C. $2 : 9 : 12$
• D. $1 : 9 : 6$

Hint:

In triangle problems with angles and sides in specific progressions, use the sine rule and trigonometric identities to relate angles, then solve using the arithmetic condition on sides.

Answer 1

The Correct Option is C

Given $a, b, c$ are in arithmetic progression, $2b = a + c$. Also, $\angle A = 2 \angle C$, and since $A + B + C = 180^\circ$, we have $2C + B + C = 180^\circ$, so $B = 180^\circ - 3C$. Use the sine rule: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \] Thus, $a = 2R \sin A = 2R \sin 2C$, $b = 2R \sin B = 2R \sin(180^\circ - 3C) = 2R \sin 3C$, $c = 2R \sin C$. The arithmetic progression condition gives: \[ 2 \cdot 2R \sin 3C = 2R \sin 2C + 2R \sin C \implies 2 \sin 3C = \sin 2C + \sin C \] Using $\sin 3C = 3 \sin C - 4 \sin^3 C$ and $\sin 2C = 2 \sin C \cos C$: \[ 2 (3 \sin C - 4 \sin^3 C) = 2 \sin C \cos C + \sin C \] Divide through by $\sin C$ (since $\sin C \neq 0$): \[ 6 - 8 \sin^2 C = 2 \cos C + 1 \implies 6 - 8 (1 - \cos^2 C) = 2 \cos C + 1 \] \[ 8 \cos^2 C - 2 \cos C - 3 = 0 \] Solve the quadratic equation for $\cos C$: \[ \cos C = \frac{2 \pm \sqrt{4 + 96}}{16} = \frac{2 \pm 10}{16} \implies \cos C = \frac{3}{4} \text{ or } \cos C = -\frac{1}{2} \] Since $C$ is an angle in a triangle, $0^\circ<C<90^\circ$, so $\cos C = \frac{3}{4}$. Then: - $\cos A = \cos 2C = 2 \cos^2 C - 1 = 2 \cdot \left(\frac{3}{4}\right)^2 - 1 = 2 \cdot \frac{9}{16} - 1 = \frac{1}{8}$ - $\cos B = \cos(180^\circ - 3C) = -\cos 3C$. Use $\cos 3C = 4 \cos^3 C - 3 \cos C$: \[ \cos 3C = 4 \cdot \left(\frac{3}{4}\right)^3 - 3 \cdot \frac{3}{4} = 4 \cdot \frac{27}{64} - \frac{9}{4} = \frac{27}{16} - \frac{36}{16} = -\frac{9}{16} \] \[ \cos B = -\left(-\frac{9}{16}\right) = \frac{9}{16} \] - $\cos C = \frac{3}{4} = \frac{12}{16}$ The ratio is: \[ \cos A : \cos B : \cos C = \frac{1}{8} : \frac{9}{16} : \frac{12}{16} \] Multiply through by 16: \[ 2 : 9 : 12 \] Option (3) is correct. Options (1), (2), and (4) do not match the computed ratio.