Sketch the graph of one cycle of tan(x)(pi/2=1.5708) and give the requested information

Domain:
Range:
Period(wavelength):
Asymptotes:

To sketch the graph of one cycle of tan(x), we need to understand the properties of the tangent function. Here's how you can get the requested information step by step:

1. Domain:
The domain of tan(x) is all real numbers except for the values where the function is undefined. Tangent is undefined at odd multiples of π/2 (1.5708), so the domain is given by the set of real numbers excluding values such as x = (2n + 1)π/2, where n is an integer. Therefore, the domain for tan(x) is (-∞, (2n + 1)π/2) U ((2n + 1)π/2, ∞), where n is an integer.

2. Range:
The range of tan(x) is all real numbers. As x approaches odd multiples of π/2 (1.5708) from the left or right, tan(x) approaches positive and negative infinity, respectively.

3. Period (wavelength):
The period of tan(x) is π radians (180 degrees). The tangent function completes one full cycle every π radians, meaning that it repeats after every π radians.

4. Asymptotes:
The graph of tan(x) has vertical asymptotes at odd multiples of π/2 (1.5708). This is because tan(x) is undefined at those points. The equation for the vertical asymptotes is x = (2n + 1)π/2, where n is an integer.

Now, let's sketch the graph:

Start by plotting the vertical asymptotes. Draw vertical lines at x = π/2, x = -π/2, x = 3π/2, x = -3π/2, and so on. These lines indicate the points where the graph is undefined.

Next, identify the key characteristic points of the graph:
- At x = 0, the tangent function is equal to 0.
- At x = π/4 and x = -π/4, the tangent function is equal to 1.
- At x = 3π/4 and x = -3π/4, the tangent function is equal to -1.

Using these points and the knowledge that the graph repeats every π radians, sketch a smooth curve that passes through these points.

Make sure the graph approaches the asymptotes as it moves away from them in both the positive and negative directions.

Your final graph should resemble a wave-like pattern, oscillating between vertical asymptotes and covering one full cycle within π radians.

Note: If you are using graphing software or a calculator, you can simply plot the function y = tan(x) over the desired interval to obtain a precise graph.