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Chapter 4: Problem 17

State the period of each function. $$y=\cot \pi x$$

Short Answer

Expert verified
Therefore, the period of the function \(y=\cot \pi x\) is 1.

Step by step solution

01

Identify the basic form

First, recognize that the given function is a cotangent function, which has a basic form \(y = \cot x\). The standard period of the cotangent function is \(\pi\). This is the interval length over which the function's values repeat.
02

Note the effect of the multiplier

The given function is not the standard cotangent function, but instead \(y = \cot \pi x\). The argument of the cotangent function has been multiplied by \(\pi\). In a general sense, when you have a sine, cosine, or cotangent function (and other trigonometric functions as well) and the argument is multiplied by a constant, that constant changes the period of the function. For \(\cot bx\), the period is \(\pi/b\). Therefore, \(\pi\) is the b-value in this function.
03

Apply the formula for the period

Apply the formula for the period of a cotangent function \(y = \cot bx\), which is \(\pi/b\). Substituting \(\pi\) for \(b\) gives us period = \(\pi/\pi = 1\).

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