Question

The triangle $P Q R$ of which the angles $P, Q R$ 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 $\frac{p}{\sin P}=\frac{q}{\sin Q}$
$=\frac{r}{\sin R}=\lambda$
$\because \cos P=\frac{\sin Q}{2 \sin R}$
$\Rightarrow \frac{q^{2}+r^{2}-p^{2}}{2 q r}=\frac{q}{2 r}$
$\Rightarrow q^{2}+r^{2}-p^{2}=q^{2}$
$\Rightarrow r^{2}=p^{2}$
$\Rightarrow r=p$
$\therefore $ Triangle is isosceles.