Understanding the Tangent Function and its Behavior with Negative Angles

tan(-x)

The tangent function (tan) is a trigonometric function that relates the ratio of the opposite side to the adjacent side of a right triangle

The tangent function (tan) is a trigonometric function that relates the ratio of the opposite side to the adjacent side of a right triangle. However, the tangent function can also be defined for any angle (not just right angles) using the unit circle.

The tangent function is defined as:

tan(x) = sin(x) / cos(x),

where sin(x) is the sine function and cos(x) is the cosine function.

Now, let’s consider tan(-x). When we have a negative angle, it means the angle is measured clockwise instead of counterclockwise.

Since the tangent function is defined as the ratio of sin(x) / cos(x), let’s look at what happens when we substitute -x into the tangent function:

tan(-x) = sin(-x) / cos(-x).

The sine function is an odd function, meaning that sin(-x) = -sin(x). So, we can rewrite the expression as:

tan(-x) = -sin(x) / cos(-x).

The cosine function is an even function, meaning that cos(-x) = cos(x). Substituting this into the expression, we get:

tan(-x) = -sin(x) / cos(x).

So, to summarize, the tangent of a negative angle is equal to the negative tangent of the positive angle. This means that the sign of the tangent function will be flipped when dealing with negative angles.

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