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Chapter 5: Problem 65

Without drawing a graph, describe the behavior of the basic tangent curve.

Short Answer

Expert verified
The basic tangent curve has a domain of all real numbers except \(\frac{(2n+1)蟺}{2}\), a range of all real numbers, a period of pi, is symmetric about the origin, and has vertical asymptotes at \(x = \frac{(2n+1)蟺}{2}\).

Step by step solution

01

Understand the Basic Tangent Curve

A tangent function is a trigonometric function defined as the ratio of the y-coordinate to the x-coordinate of a point on the unit circle. In other words, \(tan(x) = \frac{sin(x)}{cos(x)}\).
02

Determine the Domain

A tangent function is undefined wherever the cosine function is 0, and the cosine function equals zero at \(\frac{(2n+1)蟺}{2}\), where n is any integer. This means the domain of \(tan(x)\) consists of all real numbers, except \(\frac{(2n+1)蟺}{2}\).
03

Specify the Range

Since sine and cosine can take any value from -1 to 1, their ratio can take any real number. This means the range for \(tan(x)\) is all real numbers.
04

Understand the Period and Symmetry

The tangent function repeats every pi units, and is thus said to have a period of pi. The tangent function is also odd, which means it has rotational symmetry about the origin.
05

Point Out the Asymptotes

The line \(x = \frac{(2n+1)蟺}{2}\) is a vertical asymptote for the tangent function because \(tan(x)\) is undefined at these values of x.

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