The number of triangles in which tan A + tan | vedclass

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

Question

(C) Given the equation: $	an A + 	an B + 	an C = \cot A + \cot B + \cot C$. We know that for any triangle,$A + B + C = 180^{\circ}$. If we assume t

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(C) Given the equation: $	an A + 	an B + 	an C = \cot A + \cot B + \cot C$. We know that for any triangle,$A + B + C = 180^{\circ}$. If we assume the triangle is equilateral,$A = B = C = 60^{\circ}$,then $	an 60^{\circ} = \sqrt{3}$ and $\cot 60^{\circ} = \frac{1}{\sqrt{3}}$. $3\sqrt{3} 
eq \frac{3}{\sqrt{3}} = \sqrt{3}$. For a general triangle,the condition $	an A + 	an B + 	an C = \cot A + \cot B + \cot C$ implies $	an A + 	an B + 	an C = \frac{1}{	an A} + \frac{1}{	an B} + \frac{1}{	an C}$. This equation is satisfied if $A=B=C=60^{\circ}$ is not true,but rather if the angles satisfy specific constraints. However,in a triangle,$	an A + 	an B + 	an C = 	an A 	an B 	an C$. Also,$\cot A \cot B + \cot B \cot C + \cot C \cot A = 1$. It can be shown that for any triangle,the given condition leads to a contradiction or is impossible for real angles $A, B, C$ summing to $180^{\circ}$ except in degenerate cases. Thus,the number of such triangles is $0$.

The number of triangles in which $\tan A + \tan B + \tan C = \cot A + \cot B + \cot C$ is:

Similar Questions

In $\Delta ABC$,${a^2}({\cos ^2}B - {\cos ^2}C) + {b^2}({\cos ^2}C - {\cos ^2}A) + {c^2}({\cos ^2}A - {\cos ^2}B) = $

The common roots of the equations $2\sin^2 x + \sin^2 2x = 2$ and $\sin 2x + \cos 2x = \tan x$ are

In $\triangle ABC$,if $\cos A \cos B + \sin A \sin B \sin C = 1$ and $C = \frac{\pi}{2}$,then $A : B =$

In $\triangle ABC$,if $A, B, C$ are in arithmetic progression,$\Delta = \frac{\sqrt{3}}{2}$ and $r_1 r_2 = r_3 r$,then $R =$

In $\triangle ABC$,if $a, b, c$ are $5, 12$ and $13$ respectively,then $b^2 \sin 2C + c^2 \sin 2B =$

Vedclass Test Series

Exam Paper Generator

Online Exam Module