Understanding the Tangent Function | Evaluating tan(-x) and the Tangent of Negative Angles

tan(-x)

The tangent function is defined as the ratio of the sine of an angle to the cosine of that angle

The tangent function is defined as the ratio of the sine of an angle to the cosine of that angle. In trigonometry, we often study the tangent function in the context of a right triangle. For a right triangle with an angle x, the tangent of that angle is defined as the length of the side opposite to the angle divided by the length of the side adjacent to the angle.

Now, let’s consider the expression tan(-x). The negative sign in front of x indicates that we are considering the angle -x. Since the tangent function is periodic with a period of π (180 degrees), the tangent function of -x will be the same as the tangent function of x, with the sign flipped.

In other words, tan(-x) = -tan(x).

To understand why this is the case, let’s consider the unit circle. In the unit circle, the x-coordinate represents the cosine of an angle, and the y-coordinate represents the sine of an angle. Given that the tangent function is the ratio of sine to cosine, we can see that the tangent of an angle x would be the y-coordinate divided by the x-coordinate. As we move from the angle x to -x, the y-coordinate remains the same, but the x-coordinate becomes negative. Therefore, the ratio of y-coordinate to x-coordinate changes sign, resulting in -tan(x).

In summary, tan(-x) = -tan(x) is a rule of the tangent function that indicates that evaluating the tangent of a negative angle will give the negative of the tangent of the positive angle.

More Answers: