Problem 65 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \(tan(x) = \frac{sin(x)}{cos(x)}\).

Step 2: Determine the Domain

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

Step 3: 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.

Step 4: 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.

Step 5: Point Out the Asymptotes

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