## tangent (TAN)

### In mathematics, the tangent (TAN) is a trigonometric function that relates the ratio of the opposite side to the adjacent side in a right-angled triangle

In mathematics, the tangent (TAN) is a trigonometric function that relates the ratio of the opposite side to the adjacent side in a right-angled triangle. It is commonly abbreviated as “tan”.

To understand the tangent function, let’s consider a right-angled triangle ABC, where angle A is the right angle. If we take angle B as our reference angle and consider side BC as the side adjacent to angle B and side AB as the side opposite angle B, then the tangent of angle B is given by:

tan(B) = AB/BC

The tangent function can also be defined as the ratio of the sine of an angle to its cosine:

tan(B) = sin(B)/cos(B)

It is important to note that the tangent function is undefined for certain angles, specifically those angles where the adjacent side is equal to zero (e.g., angle 90 degrees or angle 270 degrees). This is because division by zero is undefined in mathematics.

The tangent function has a periodic nature, which means it repeats its values at regular intervals. It has a period of π radians or 180 degrees, meaning that the value of tangent for an angle is the same as the value of tangent for that angle plus or minus multiples of π radians or 180 degrees. The tangent function also has vertical asymptotes at these points where it is undefined.

In addition to its trigonometric significance, the tangent function finds applications in many areas of mathematics, science, and engineering. It is particularly useful in solving problems involving angles and triangles, as well as in analyzing periodic phenomena such as waves and oscillations.

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