What does the tangent function equal for a negative angle?

The tangent function for an angle, specifically a negative angle, is defined in terms of the sine and cosine functions. The tangent of an angle ( x ) is given by the ratio of the sine and cosine of that angle:

[

\tan(x) = \frac{\sin(x)}{\cos(x)}

]

When considering a negative angle, ( -x ), we can use the properties of sine and cosine. The sine function is odd, meaning:

[

\sin(-x) = -\sin(x)

]

The cosine function is even, meaning:

[

\cos(-x) = \cos(x)

]

Using these properties, the tangent of a negative angle can be expressed as follows:

[

\tan(-x) = \frac{\sin(-x)}{\cos(-x)} = \frac{-\sin(x)}{\cos(x)} = -\frac{\sin(x)}{\cos(x)} = -\tan(x)

]

This shows that the tangent of a negative angle, ( -x ), is equal to the negative of the tangent of the positive angle ( x ). This supports the correctness of the assertion that the tangent function for a negative angle equals (-\