Question

Which function’s graph has asymptotes located at the values x=+-nπ?

Please explain. Thank you!

Answer 1

at what degrees on the unit circle, is the line vertical

Answer 2

look at the graph of tangent, it is the opposite side over the adjacent side of a triangle, at 90 degrees or pi/2 on the unit circle, the tangent is undefined

Answer 3

the cotangent is 1/tangent, or adjacent side over opposite side, the opposite size is zero at intervals of pi degrees, and thus the cotangent is undefined and has vertical asymptopes at intervals of pi

Answer 4

so the cotangent is one of the answers

Answer 5

well crap, that gives it away, cuz only one answer involves the cotangent

Answer 6

and then obviously cosecant would be the other one making that D.

Answer 7

think about it this way
you'd get an asymptote, when a fraction has a denominator of 0
so \(\bf tan(\theta)=\cfrac{opposite}{adjacent}=\cfrac{y}{x}=\cfrac{{\color{brown}{ sin}}(\theta)}{{\color{blue}{ cos}}(\theta)}

\\ \quad \\


csc(\theta)=\cfrac{hypotenuse}{opposite}=\cfrac{r}{y}=\cfrac{1}{{\color{brown}{ sin}}(\theta)}
\\ \quad \\
sec(\theta)=\cfrac{hypotenuse}{adjacent}=\cfrac{r}{x}=\cfrac{1}{{\color{blue}{ cos}}(\theta)}
\\ \quad \\
cot(\theta)=\cfrac{adjacent}{opposite}=\cfrac{y}{x}=\cfrac{{\color{blue}{ cos}}(\theta)}{{\color{brown}{ sin}}(\theta)}\)

if you use the angle \(\Large \pi\) there for all
which ones yield a 0 at the bottom?

the ones that do, have an asymptote at that angle

Answer 8

CSC x = 1 / sin x , sin x = 0 at intervals of pi also, the opposite side of the triangle is zero on the unit circle at intervals of pi degrees.

Therefore, CSC is undefined and has vertical asmytopes at intervals of pi degrees also

Answer 9

I want to thank you both for your explanations because I want understand why and not just the answer!

Answer 10

The unit circle is your best friend, just imagine that thing, and all the answers are there

Answer 11

yw