Question

The triangle $PQR$ of which the angles $P,QR$ satisfy $\cos P=\frac{\sin Q}{2\sin R}$ is

• A. equilateral
• B. right angled
• C. any triangle
• D. isosceles

Answer 1

Answer: (D)

Let sides of a triangle are $p,q,r$.
Applying sine rule $\frac{p}{\sin P}=\frac{q}{\sin Q}$
$=\frac{r}{\sin R}=\lambda$
$\because \cos P=\frac{\sin Q}{2\sin R}$
$\Rightarrow \frac{q^2+r^2-p^2}{2qr}=\frac{q}{2r}$
$\Rightarrow q^2+r^2-p^2=q^2$
$\Rightarrow r^2=p^2$
$\Rightarrow r=p$
$\therefore$ Triangle is isosceles.