Question

The area of triangle $ABC$ is equal to $\left( {{a^2} + {b^2} - {c^2}} \right)$ where $a$, $b$ and $c$ are the sides of triangle. Find the value of $\tan C$.

Answer 1

Here we need to find the tangent of angle of the triangle. We will first draw the triangle with their sides and angles and then we will use the formula of area triangle and then the formula of sine of angle of the triangle in terms of the area and then we will find the cosine of angle of the triangle in terms of the area and to get the tangent of angle of the triangle, we will take the ratio of both of them.

Formula used:
$\Delta = \dfrac{1}{2}ab\sin C$, where, $a$ and $b$ are the length of the sides of the triangle and $\Delta$ is the area of the triangle.

Complete step-by-step answer:
We will first draw the triangle $ABC$, $a$, $b$ and $c$ are the sides of the triangle.

It is given that
$\Delta = \left( {{a^2} + {b^2} - {c^2}} \right)$ …………. $\left( 1 \right)$
We know that the area of triangle in terms of sides and angle of the triangle is given by
$\Rightarrow \Delta = \dfrac{1}{2}ab\sin C$
On further simplification, we get
$\Rightarrow \sin C = \dfrac{{2\Delta }}{{ab}}$ ………………… $\left( 2 \right)$
And we also know that $\cos C = \dfrac{{{a^2} + {b^2} - {c^2}}}{{2ab}}$.
On substituting equation $\left( 1 \right)$ in the above equation, we get
$\Rightarrow \cos C = \dfrac{\Delta }{{2ab}}$ ……………. $\left( 3 \right)$
Now, we will take the ratio of the equation $\left( 2 \right)$ and equation $\left( 3 \right)$. Therefore, we get
$\dfrac{{\sin C}}{{\cos C}} = \dfrac{{\dfrac{{2\Delta }}{{ab}}}}{{\dfrac{\Delta }{{2ab}}}}$
Using the trigonometric formula $\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta$, we get
$\Rightarrow \tan C = \dfrac{{\dfrac{{2\Delta }}{{ab}}}}{{\dfrac{\Delta }{{2ab}}}}$
On further simplifying the fraction of the right hand side of the equation.
$\Rightarrow \tan C = 2 \times 2 = 4$
Hence, the value of $\tan C$ is equal to 4.

Note: A triangle is a two dimensional figure which has three sides. We have also used basic trigonometric identities to find different angles of the triangle. Trigonometric identities are defined as the equalities that contain the trigonometric function and also it is true for all the values of the variables.