Question

Which of the following are vertical asymptotes of the function y=3cot(2x) - 4? Check all that apply.

Answer 1

Given the function:

`y=3cot(2x)-4`

Let's find the vertical asymptotes of the function.

To find the vertical asymptote, set 2x equal to zero:

`2x=0`

Solve for x.

To solve, divide both sides by 2:

`\begin{gathered} (2x)/(2)=(0)/(2) \\ \\ x=0 \end{gathered}`

Also, term inside the cotangent function, (2x), to π:

`2x=\pi`

Solve for x by dividing both sides by 2:

`\begin{gathered} (2x)/(2)=(\pi)/(2) \\ \\ x=(\pi)/(2) \end{gathered}`

Also, apply the trigonometric identity:

`cot=(cos)/(sin)`

Hence, the vertical asymptote will be where sin(2x) equals 0.

Now, input all given choices for x in sin(2x) and solve:

`\begin{gathered} sin(2*(\pi)/(3))=0.86 \\ \\ sin(2*(\pi)/(2))=sin\pi=0 \\ \\ sin(2*2\pi)=sin4\pi=0 \\ \\ sin(\pi)=0 \end{gathered}`

Therefore, the vertical asymptotes of the given function are:

`\begin{gathered} x=\pm(\pi)/(2) \\ \\ x=2\pi \\ \\ x=\pi \end{gathered}`

ANSWER:

B. x = ±π/2

C. x = 2π

D. x = π