Question

In a $\Delta ABC,\cos A+\cos B+\cos C>1$ only if the triangle is

• A. acute angled
• B. obtuse angled
• C. right angled
• D. the nature of the triangle cannot be determined

Answer 1

The correct option is D the nature of the triangle cannot be determined

We know in a $\Delta ABC$

$\cos A+\cos B+\cos C=1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

Now given $\cos A+\cos B+\cos C>1$

$\Rightarrow 1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}>1$

$\Rightarrow \sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}>0$

But this condition is true for all types of triangle.

Hence we can't determined the nature of triangle from this.