Understanding the Tangent Function: Formula, Calculation, and Periodicity

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The tangent function, commonly denoted as tan(x), is one of the six trigonometric functions and represents the ratio of the length of the side opposite an angle in a right triangle to the length of the adjacent side

The tangent function, commonly denoted as tan(x), is one of the six trigonometric functions and represents the ratio of the length of the side opposite an angle in a right triangle to the length of the adjacent side. In other words, tan(x) is equal to the opposite side divided by the adjacent side.

To understand the concept of tangent, it is important to recall the basic trigonometric ratios in a right triangle. In a right triangle, there are three sides: the hypotenuse (the longest side), the opposite side (the side opposite to the angle you are interested in), and the adjacent side (the side adjacent to the angle). The tangent of an angle is found by dividing the length of the opposite side by the length of the adjacent side.

Mathematically, the tangent of an angle x is given by the formula:

tan(x) = opposite/adjacent

For example, let’s consider a right triangle with an angle x and sides of lengths opposite and adjacent. If we want to find the value of tan(x), we divide the length of the side opposite angle x by the length of the side adjacent to angle x.

For instance, if the opposite side has a length of 3 units and the adjacent side has a length of 4 units, we can calculate the value of tan(x) as follows:

tan(x) = opposite/adjacent
tan(x) = 3/4
tan(x) = 0.75

Hence, in this case, the value of tan(x) is 0.75.

It is important to note that the tangent function is periodic with a period of π radians or 180 degrees. This means that if you increase the angle by multiples of π or 180 degrees, the values of the tangent function will repeat. Additionally, the tangent function is undefined at certain angles, such as when the adjacent side is equal to 0 (since division by zero is undefined).

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