tan(-x)
The tangent function, denoted as tan(x), is a trigonometric function that relates the ratio of the opposite side of a right triangle to the adjacent side
The tangent function, denoted as tan(x), is a trigonometric function that relates the ratio of the opposite side of a right triangle to the adjacent side. The input of the function, x, represents an angle in radians.
In the case of tan(-x), the negative sign in front of x reflects a negative angle. This means that the angle is measured clockwise from the positive x-axis instead of counterclockwise.
To understand the value of tan(-x), let’s consider the unit circle.
The unit circle is a circle with a radius of 1 centered at the origin (0,0) on the Cartesian plane. The positive x-axis is along the right side of the circle, and the positive y-axis is along the top of the circle.
For positive angles, the ratio of the opposite side to the adjacent side is positive, while for negative angles, the ratio is negative.
Since tan(x) = sin(x) / cos(x), we can consider the ratios of sine and cosine.
For any angle x on the unit circle, the opposite side is represented by the y-coordinate and the adjacent side is represented by the x-coordinate.
When we have a negative angle -x, it means that we need to start from the positive x-axis and rotate clockwise by an angle of x. So, we have to consider the coordinates on the unit circle.
Let’s consider an example where x = π/4. In this case, the angle in the counterclockwise direction is π/4, and the angle in the clockwise direction is -π/4.
If we draw a line from the origin on the unit circle to the point (cos(-π/4), sin(-π/4)), we find that cos(-π/4) = cos(π/4) and sin(-π/4) = -sin(π/4).
Therefore, tan(-π/4) = sin(-π/4) / cos(-π/4) = -sin(π/4) / cos(π/4) = -tan(π/4).
In general, tan(-x) = -tan(x) for any angle x. The negative sign in front of the tangent function indicates that the ratio of the opposite side to the adjacent side is negative.
So, if you have a problem that requires finding the value of tan(-x), you can calculate tan(x) and negate the result.
Keep in mind that if the angle x is a multiple of π, tan(x) may not be defined or may approach infinity. In such cases, tan(-x) will also be undefined or approach negative infinity.
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