If in a triangle ABC,cos A + cos B + cos C = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) We know that for any triangle $ABC$,the maximum value of $\cos A + \cos B + \cos C$ is $\frac{3}{2}$,which occurs if and only if $A = B = C = 60^{

Vedclass · Accepted Answer

Equilateral

Vedclass · Answer

(B) We know that for any triangle $ABC$,the maximum value of $\cos A + \cos B + \cos C$ is $\frac{3}{2}$,which occurs if and only if $A = B = C = 60^{\circ}$.Alternatively,using the identity $\cos A + \cos B + \cos C = 1 + 4 \sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2}) = \frac{3}{2}$,we get $\sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2}) = \frac{1}{8}$.Since the maximum value of $\sin(\frac{A}{2}) \sin(\frac{B}{2}) \sin(\frac{C}{2})$ is $\frac{1}{8}$ (by Jensen's inequality for concave functions),this equality holds only when $\frac{A}{2} = \frac{B}{2} = \frac{C}{2} = 30^{\circ}$,i.e.,$A = B = C = 60^{\circ}$.Thus,the triangle is equilateral.

If in a triangle $ABC$,$\cos A + \cos B + \cos C = \frac{3}{2}$,then the triangle is

Similar Questions

In a triangle $ABC$,if $b=7, c=4\sqrt{3}$ and $A=\frac{\pi}{6}$,then $a \sin B \sin C =$

In $\triangle ABC$,if $A: B: C = 5: 1: 6$,then $a: b: c =$

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. If the triangle $ADC$ is equilateral,then the ratio $a^2 : b^2 : c^2$ is equal to

If in a triangle $\left(1-\frac{r_1}{r_2}\right)\left(1-\frac{r_1}{r_3}\right)=2$,then the triangle is

In triangle $ABC$,if $BC$ is the hypotenuse,then $r_2 + r_3 =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module