Understanding the Tangent Function: Definition, Properties, and Applications

tan(u)

The tangent function, denoted as tan(u), is a trigonometric function that relates the ratio of the length of the opposite side to the adjacent side of a right triangle

The tangent function, denoted as tan(u), is a trigonometric function that relates the ratio of the length 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 that angle.

Mathematically, the tangent function can be defined as:

tan(u) = sin(u) / cos(u)

The tangent function is periodic and has a repeating pattern every π radians (180 degrees). It has horizontal asymptotes at multiples of π/2 radians (90 degrees), where the function becomes undefined.

To find the value of tan(u), you will need to know the value of the angle u in radians or degrees. If you have the angle in degrees, you can convert it to radians by multiplying it by π/180.

For example, if u = 45 degrees, the equivalent angle in radians would be u = (45 * π)/180 = π/4 radians.

To evaluate the tan(u) using a calculator, you can directly input the angle in radians or degrees and calculate the tangent function.

For instance, if your calculator is set to degrees mode and you want to find tan(45), you would enter “45” and press the tangent button (usually abbreviated as “tan” or represented by a “T”).

If you are working with a right triangle and have the lengths of the sides, you can also use the tangent function to find the value of an angle. In this case, you would take the inverse tangent (arctan or tan^(-1)) of the ratio of the lengths of the opposite and adjacent sides.

It is important to keep in mind that some angles may result in an undefined or infinite value for the tangent function. For example, tan(90 degrees) is undefined as the cosine of 90 degrees is zero.

Overall, the tangent function is a fundamental trigonometric function that is frequently used in various applications in mathematics, physics, and engineering.

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