# Tan of negative angle

A mathematical relation of tangent of negative angle with tangent of positive angle is called tan of negative angle identity.

## Formula

$\tan{(-\theta)} \,=\, -\tan{\theta}$

### Proof

The mathematical relation between tangent of negative angle and tan of positive angle is derived in mathematical form by geometrical method in trigonometry.

#### Construction of triangle with positive angle

$\Delta POQ$ is a right angled triangle and take its positive angle is denoted by theta. Express tan of positive angle in terms of ratio of lengths of the two respective sides.

$\tan{\theta} \,=\, \dfrac{PQ}{OQ}$

The $\Delta POQ$ is constructed in first quadrant. Therefore, the lengths of both adjacent and opposite sides are positive and take they both are represented by $x$ and $y$ respectively.

$\implies$ $\tan{\theta} \,=\, \dfrac{y}{x}$

#### Construction of triangle with negative angle

Construct the same triangle with negative angle. Therefore, the angle of $\Delta QOR$ is negative theta, denoted by $–\theta$.

On the basis of this data, write the tan of negative angle in terms of ratio of lengths of respective sides.

$\tan{(-\theta)} \,=\, \dfrac{QR}{OQ}$

Due to construction of the triangle with negative angle, geometrically the length of opposite side will be $–y$ but the length of adjacent side is same.

$\implies$ $\tan{(-\theta)} \,=\, \dfrac{-y}{x}$

#### Comparing Cosine functions

Now, compare equations of both tangent of positive angle and tan of negative angle.

$\tan{\theta} \,=\, \dfrac{y}{x}$

$\tan{(-\theta)} \,=\, -\dfrac{y}{x}$

The two equations disclose that tan of negative angle is always equal to negative of tangent of angle.

$\,\,\, \therefore \,\,\,\,\,\,$ $\tan{(-\theta)} \,=\, -\tan{\theta}$

It is called tangent of negative angle identity and used as a formula in trigonometric mathematics.

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Jun 26, 2023

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