Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
g(θ)=tan πθ/10

Verified step by step guidance

1. Step 1: Recognize that the function g(θ) = tan(πθ/10) is a tangent function, which has vertical asymptotes where its argument is an odd multiple of π/2.
2. Step 2: Set the argument of the tangent function equal to (2n+1)π/2, where n is an integer, to find the vertical asymptotes: πθ/10 = (2n+1)π/2.
3. Step 3: Solve the equation from Step 2 for θ by multiplying both sides by 10/π to isolate θ: θ = (2n+1) * 5.
4. Step 4: Simplify the expression for θ to find the general form of the vertical asymptotes: θ = 5(2n+1).
5. Step 5: Conclude that the vertical asymptotes occur at θ = 5, 15, 25, ..., for n = 0, 1, 2, ..., and at θ = -5, -15, -25, ..., for n = -1, -2, -3, ..., indicating the positions of the vertical asymptotes.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. This typically happens at points where the function is undefined, such as where the denominator of a rational function equals zero. For trigonometric functions like tangent, vertical asymptotes can also occur at specific angles where the function is undefined.

Tangent Function

The tangent function, denoted as tan(θ), is a periodic function defined as the ratio of the sine and cosine functions: tan(θ) = sin(θ)/cos(θ). It has a period of π, meaning it repeats its values every π radians. The function is undefined wherever the cosine function equals zero, which leads to vertical asymptotes in its graph.

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and their properties. By plotting functions, users can easily identify key features such as intercepts, asymptotes, and behavior at infinity. For functions like g(θ) = tan(πθ/10), a graphing utility can help quickly locate vertical asymptotes by showing where the function approaches infinity.