Browse Questions

# For a $\Delta ABC$ it is given that : $\cos A+\cos B+\cos C=\large\frac{3}{2}$.What type of triangle is $ABC$?

$\begin{array}{1 1}(a)\;Isosceles \;triangle&(b)\;Right\;angled\;triangle\\(c)\;equilateral\;triangle&(d)\;None\;of\;these\end{array}$

Step 1:
Given: $\cos A+\cos B+\cos C=\large\frac{3}{2}$
$\Rightarrow \large\frac{b^2+c^2-a^2}{2bc}+\frac{c^2+a^2-b^2}{2ca}+\frac{a^2+b^2-c^2}{2ab}=\frac{3}{2}$
$\Rightarrow a(b^2+c^2-a^2)+b(c^2+a^2-b^2)+c(a^2+b^2-c^2)=3abc$
$\Rightarrow a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)=a^3+b^3+c^3+3abc$
$\Rightarrow a(b^2+c^2-2bc)+b(c^2+a^2-2ac)+c(a^2+b^2-2ab)=a^3+b^3+c^3-3abc$