Solved on Feb 22, 2024

Find the true statement about the cotangent function $y=\cot x$ .
A. Zeros are $\frac{n \pi}{2}$ where $n$ is an integer. B. Domain excludes $x$ where $\cos x=0$ . C. Principal cycle is on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ . D. Infinitely many vertical asymptotes at $x=n \pi$ where $n$ is an integer.

STEP 1

Assumptions
1. The function given is $y = \cot x$ .
2. We need to evaluate the truthfulness of each statement regarding the properties of the cotangent function.

STEP 2

Evaluate statement A: The zeros of $y=\cot x$ are of the form $\frac{n \pi}{2}$ where $n$ is an integer.
The cotangent function, $\cot x$ , is defined as $\cot x = \frac{\cos x}{\sin x}$ . The zeros of the cotangent function occur when the numerator, $\cos x$ , equals zero, which is not correct. The zeros occur when the denominator, $\sin x$ , equals zero. Therefore, we need to find the values of $x$ for which $\sin x = 0$ .

STEP 3

Solve for the zeros of $\sin x$ .
The $\sin x$ function has zeros at $x = n\pi$ where $n$ is an integer. This is because the sine function is zero at integer multiples of $\pi$ .

STEP 4

Compare the zeros of $\sin x$ with the statement A.
Since the zeros of $\sin x$ are at $x = n\pi$ , and not at $\frac{n \pi}{2}$ , statement A is false.

STEP 5

Evaluate statement B: The domain of $y=\cot x$ is all real numbers except for values of $x$ for which $\cos x=0$ .
The domain of the cotangent function is all real numbers except for the values that make the denominator of $\cot x = \frac{\cos x}{\sin x}$ zero, which are the values where $\sin x = 0$ .

STEP 6

Identify the values of $x$ for which $\sin x = 0$ .
As previously stated, $\sin x = 0$ at $x = n\pi$ where $n$ is an integer.

STEP 7

Compare the domain restriction of $\cot x$ with statement B.
Since the domain of $y=\cot x$ is restricted by the values of $x$ where $\sin x = 0$ , which are $x = n\pi$ , statement B is true.

STEP 8

Evaluate statement C: The principal cycle of the graph of $y=\operatorname{cotx}$ occurs on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ .
The cotangent function has a period of $\pi$ , and its principal cycle occurs where the function is defined and not interrupted by vertical asymptotes. The vertical asymptotes of $\cot x$ occur where $\sin x = 0$ , which is at $x = n\pi$ .

STEP 9

Identify the interval of the principal cycle of $y=\cot x$ .
The principal cycle of $\cot x$ is the interval between two consecutive asymptotes, which is $(0, \pi)$ or equivalently $(-\pi, 0)$ . This interval does not match the interval given in statement C.

STEP 10

Compare the principal cycle interval with statement C.
Since the principal cycle of $\cot x$ is not on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , statement C is false.

STEP 11

Evaluate statement D: The function $y=\cot x$ has infinitely many vertical asymptotes with equations $x=n \pi$ where $n$ is an integer.
As previously mentioned, the vertical asymptotes of $\cot x$ occur where $\sin x = 0$ , which is at $x = n\pi$ .

STEP 12

Confirm the presence of vertical asymptotes for $y=\cot x$ .
Since $\cot x$ has vertical asymptotes at every integer multiple of $\pi$ , statement D is true.

STEP 13

Determine the correct statement.
Based on the evaluations, statement B and statement D are true.
The correct answer is B and D.

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STEP 1

Assumptions
1. The function given is $y = \cot x$ .
2. We need to evaluate the truthfulness of each statement regarding the properties of the cotangent function.

STEP 2

STEP 3

Solve for the zeros of $\sin x$ .
The $\sin x$ function has zeros at $x = n\pi$ where $n$ is an integer. This is because the sine function is zero at integer multiples of $\pi$ .

STEP 4

Compare the zeros of $\sin x$ with the statement A.
Since the zeros of $\sin x$ are at $x = n\pi$ , and not at $\frac{n \pi}{2}$ , statement A is false.

STEP 5

STEP 6

Identify the values of $x$ for which $\sin x = 0$ .
As previously stated, $\sin x = 0$ at $x = n\pi$ where $n$ is an integer.

STEP 7

Compare the domain restriction of $\cot x$ with statement B.
Since the domain of $y=\cot x$ is restricted by the values of $x$ where $\sin x = 0$ , which are $x = n\pi$ , statement B is true.

STEP 8

STEP 9

Identify the interval of the principal cycle of $y=\cot x$ .
The principal cycle of $\cot x$ is the interval between two consecutive asymptotes, which is $(0, \pi)$ or equivalently $(-\pi, 0)$ . This interval does not match the interval given in statement C.

STEP 10

Compare the principal cycle interval with statement C.
Since the principal cycle of $\cot x$ is not on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , statement C is false.

STEP 11

Evaluate statement D: The function $y=\cot x$ has infinitely many vertical asymptotes with equations $x=n \pi$ where $n$ is an integer.
As previously mentioned, the vertical asymptotes of $\cot x$ occur where $\sin x = 0$ , which is at $x = n\pi$ .

STEP 12

Confirm the presence of vertical asymptotes for $y=\cot x$ .
Since $\cot x$ has vertical asymptotes at every integer multiple of $\pi$ , statement D is true.

STEP 13

Determine the correct statement.
Based on the evaluations, statement B and statement D are true.
The correct answer is B and D.

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STEP 1

Assumptions1. The function given is y=cot⁡xy = \cot xy=cotx.2. We need to evaluate the truthfulness of each statement regarding the properties of the cotangent function.

STEP 2

STEP 3

Solve for the zeros of sin⁡x\sin xsinx.The sin⁡x\sin xsinx function has zeros at x=nπx = n\pix=nπ where nnn is an integer. This is because the sine function is zero at integer multiples of π\piπ.

STEP 4

Compare the zeros of sin⁡x\sin xsinx with the statement A.Since the zeros of sin⁡x\sin xsinx are at x=nπx = n\pix=nπ, and not at nπ2\frac{n \pi}{2}2nπ​, statement A is false.

STEP 5

STEP 6

Identify the values of xxx for which sin⁡x=0\sin x = 0sinx=0.As previously stated, sin⁡x=0\sin x = 0sinx=0 at x=nπx = n\pix=nπ where nnn is an integer.

STEP 7

Compare the domain restriction of cot⁡x\cot xcotx with statement B.Since the domain of y=cot⁡xy=\cot xy=cotx is restricted by the values of xxx where sin⁡x=0\sin x = 0sinx=0, which are x=nπx = n\pix=nπ, statement B is true.

STEP 8

STEP 9

STEP 10

Compare the principal cycle interval with statement C.Since the principal cycle of cot⁡x\cot xcotx is not on the interval (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)(−2π​,2π​), statement C is false.

STEP 11

Evaluate statement D: The function y=cot⁡xy=\cot xy=cotx has infinitely many vertical asymptotes with equations x=nπx=n \pix=nπ where nnn is an integer.As previously mentioned, the vertical asymptotes of cot⁡x\cot xcotx occur where sin⁡x=0\sin x = 0sinx=0, which is at x=nπx = n\pix=nπ.

STEP 12

Confirm the presence of vertical asymptotes for y=cot⁡xy=\cot xy=cotx.Since cot⁡x\cot xcotx has vertical asymptotes at every integer multiple of π\piπ, statement D is true.

STEP 13

Determine the correct statement.Based on the evaluations, statement B and statement D are true.The correct answer is B and D.

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Assumptions
1. The function given is $y = \cot x$ .
2. We need to evaluate the truthfulness of each statement regarding the properties of the cotangent function.

Solve for the zeros of $\sin x$ .
The $\sin x$ function has zeros at $x = n\pi$ where $n$ is an integer. This is because the sine function is zero at integer multiples of $\pi$ .

Compare the zeros of $\sin x$ with the statement A.
Since the zeros of $\sin x$ are at $x = n\pi$ , and not at $\frac{n \pi}{2}$ , statement A is false.

Identify the values of $x$ for which $\sin x = 0$ .
As previously stated, $\sin x = 0$ at $x = n\pi$ where $n$ is an integer.

Compare the domain restriction of $\cot x$ with statement B.
Since the domain of $y=\cot x$ is restricted by the values of $x$ where $\sin x = 0$ , which are $x = n\pi$ , statement B is true.

Compare the principal cycle interval with statement C.
Since the principal cycle of $\cot x$ is not on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , statement C is false.

Evaluate statement D: The function $y=\cot x$ has infinitely many vertical asymptotes with equations $x=n \pi$ where $n$ is an integer.
As previously mentioned, the vertical asymptotes of $\cot x$ occur where $\sin x = 0$ , which is at $x = n\pi$ .

Confirm the presence of vertical asymptotes for $y=\cot x$ .
Since $\cot x$ has vertical asymptotes at every integer multiple of $\pi$ , statement D is true.

Determine the correct statement.
Based on the evaluations, statement B and statement D are true.
The correct answer is B and D.