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

Hint: 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 \[{180^ \circ }\].