Question

If in a $\Delta ABC$, $\tan A + \tan B + \tan C > 0$, then,
A. Triangle is always obtuse angled triangle
B. Triangle is always equilateral triangle
C. Triangle is always acute angled triangle
D. Nothing can be said about the triangle.

Answer 1

Here we are going to use the given condition in the tangent relation of angles, by finding the relation between the angles we can come to a conclusion about the angles. From there we will identify whether the angles are acute or obtuse.

Complete step-by-step answer:
We know that the sum of all the angles of the triangle is $\pi$.
So, $\angle A + \angle B + \angle C = \pi$
We know the formula for $\tan (A + B + C)$in trigonometric identity,
That is $\tan (A + B + C) = \dfrac{{\tan A + \tan B + \tan C - \tan A\tan B\tan C}}{{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}}$
Now let us substitute $\angle A + \angle B + \angle C = \pi$ in the above given tangent relation then we get,
$\dfrac{{\tan A + \tan B + \tan C - \tan A\tan B\tan C}}{{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}} = \tan \pi$
As we know that $\tan \pi = 0$ the above relation can be written as follows,
$\dfrac{{\tan A + \tan B + \tan C - \tan A\tan B\tan C}}{{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}} = 0$
Let us multiply both sides of the equation by $1 - \tan A\tan B - \tan B\tan C - \tan C\tan A$ on both sides of the above relation then we get,
$\tan A + \tan B + \tan C - \tan A\tan B\tan C = 0$
On simplifying the above relation by adding $\tan A\tan B\tan C$ on both sides of the above relation we get,
$\tan A + \tan B + \tan C = \tan A\tan B\tan C$…… (1)
Also it is given in the problem that,
$\tan A + \tan B + \tan C > 0$
Then from the above equation (1) we have,
$\tan A\tan B\tan C > 0$
This is possible only if $\tan A > 0,\tan B > 0,\tan C > 0$ we know that the value of tangent is always positive in the first and third quadrant.
Here we would choose the first quadrant, that is $\tan A > 0, \tan B > 0, \tan C > 0$ would be possible only when three angles are greater than ${0^ \circ }$and less than${90^ \circ }$.
Hence we have found the angles of the triangle are acute.

So, the correct answer is “Option C”.

Note: An angle is said to be an acute angle is the value of the angles is greater than ${0^ \circ }$ and less than ${90^ \circ }$. If they are greater than ${90^ \circ }$ we say the angle as obtuse angles obtuse angles are always less than ${180^ \circ }$.