## 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. It is defined as the ratio of the sine of an angle to the cosine of the same angle.

In the case of tan(-x), it represents the tangent of the negative angle, where x is the angle in radians. To understand this, it’s important to know the properties of the trigonometric functions.

When dealing with angles in quadrants, the sine and cosine functions have positive and negative values based on the quadrant. In the first quadrant (0 to π/2), both sine and cosine are positive. In the second quadrant (π/2 to π), sine is positive but cosine is negative. In the third quadrant (π to 3π/2), both sine and cosine are negative. In the fourth quadrant (3π/2 to 2π), sine is negative but cosine is positive.

Now, let’s look at the tangent function. Tangent is defined as sin(x) / cos(x). By substituting the values of sine and cosine for the different quadrants, we can determine the signs of tangent.

In the first quadrant: tan(x) = sin(x) / cos(x) (both positive / positive) = positive

In the second quadrant: tan(x) = sin(x) / cos(x) (positive / negative) = negative

In the third quadrant: tan(x) = sin(x) / cos(x) (negative / negative) = positive

In the fourth quadrant: tan(x) = sin(x) / cos(x) (negative / positive) = negative

From this, we can conclude that the signs of the tangent function alternate between positive and negative for successive quadrants.

Now, getting back to tan(-x), when we introduce a negative sign to the angle x, it essentially flips the quadrant. If x was in the first quadrant, -x will be in the fourth quadrant, and vice versa. So, when we take the tangent of -x, we essentially calculate the tangent of the angle x in the opposite quadrant.

Therefore, tan(-x) will have the same numerical value as tan(x), but with the opposite sign. For example, if tan(x) = 2, then tan(-x) = -2. This is because we are essentially flipping the sign of the ratio while keeping the absolute value the same.

In summary, tan(-x) represents the tangent of the angle x in the opposite quadrant, with the same numerical value but opposite sign compared to tan(x).

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