Understanding the Tangent Function | Definition, Properties, and Applications

Tan(x)

In mathematics, the tangent function, denoted as tan(x), is a trigonometric function that relates the ratio of the length of the opposite side (O) to the adjacent side (A) of a right triangle to the angle (x) formed between the hypotenuse and the adjacent side

In mathematics, the tangent function, denoted as tan(x), is a trigonometric function that relates the ratio of the length of the opposite side (O) to the adjacent side (A) of a right triangle to the angle (x) formed between the hypotenuse and the adjacent side.

The formula for the tangent function is:

tan(x) = O / A

The tangent function is defined for all real numbers, except for the values where the cosine function is equal to zero. These values occur at x = (2n + 1)π/2, where n is an integer. At these points, the tangent function is said to be undefined because dividing by zero is not possible.

The tangent function has several key properties:

1. Periodicity: The tangent function has a periodicity of π, meaning that it repeats itself every π units. This can be observed by plotting the function on a graph.

2. Symmetry: The tangent function is an odd function, which means that it exhibits symmetry about the origin (0,0). This property can be seen from the fact that tan(-x) = -tan(x).

3. Range: The tangent function has a range of all real numbers. This means that for any given value of x, the tan(x) can take on any real value, including positive, negative, and zero.

4. Asymptotes: The tangent function has vertical asymptotes at x = (2n + 1)π/2, where n is an integer. As x approaches these values, the function approaches positive or negative infinity as the length of the adjacent side (A) gets arbitrarily close to zero.

5. Trigonometric identity: The tangent function is related to the sine and cosine functions by the identity: tan(x) = sin(x) / cos(x).

The tangent function is widely used in mathematics and various fields such as physics, engineering, and computer graphics. It helps in solving problems related to angles, trigonometry, and geometry.

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