Question

If sides of a triangle are in A.P. and the greatest angle is two times the smallest angle in the trian then cosine of least angle is

• A. 3/8
• B. 2/3
• C. 1/8
• D. 3/4

Answer 1

The correct option is D 3/4

Let angles of triangle be $A,B,C$

with corresponding sides $a,b,c$.

Let $A=\theta ,B=n-3\theta ,C=\theta$

$2b=a+c$

$=\frac{\sin C}{c}=\frac{\sin C}{c}=\frac{\sin A}{a}=k$

$\Rightarrow b=\frac{\sin B}{k},c=\frac{\sin C}{k},a=\frac{\sin A}{k}$

$\Rightarrow \frac{2\sin B}{k}=\frac{\sin A}{k}+\frac{\sin C}{k}$

$=2\sin 3\theta =\mathrm{Sin}\theta +\sin 2\theta$

$=2(3\sin \theta -4{\sin }^3\theta )=\sin \theta (1+2\cos \theta )$

$=6-8(1-{\cos }^2\theta )=1+2\cos \theta$

$\Rightarrow \cos \theta =\frac{3}{4},\frac{-1}{2}$

But $\cos \theta \ne \frac{-1}{2}$

$\therefore \cos \theta =\frac{3}{4}$