Understanding tan(-θ): A Review of the Tangent Function and Equivalent Positive Angles

tan(-θ)

To understand the value of tan(-θ), let’s first review the basic definition of the tangent function

To understand the value of tan(-θ), let’s first review the basic definition of the tangent function.

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In other words, for an angle θ, tan(θ) = opposite/adjacent.

Now, let’s consider the negative angle -θ. In trigonometry, negative angles are measured in the clockwise direction from the positive x-axis. Since the tangent function is periodic with a period of π (180 degrees), we can find the value of tan(-θ) by considering an equivalent positive angle.

To find an equivalent positive angle, we can simply add a full revolution (2π radians or 360 degrees) to the negative angle. So, tan(-θ) = tan(-θ + 2π).

Now, if we think about the unit circle, a negative angle -θ represents an angle that is clockwise from the positive x-axis, whereas the equivalent positive angle (-θ + 2π) would represent the same angle measured counterclockwise from the positive x-axis.

In both cases, the length of the side opposite to the angle and the length of the side adjacent to the angle remain the same. Therefore, the ratio of the opposite side to the adjacent side remains unchanged.

Hence, we can conclude that tan(-θ) = tan(-θ + 2π), meaning that the tangent of a negative angle is equal to the tangent of its equivalent positive angle.

In summary, tan(-θ) = tan(-θ + 2π) for any angle θ.

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