Graphs of tan x and cot x

[nycmathguy]

We are now in Section 4.6. Before graphing a few tangent and cotangent trigonometric functions, I want to say a few things and then ask a question or two.

1. The graph of y = tan x is very similar to
y = - cot x on the xy-plane. By "similar" I mean they travel from right to left in a curve-like manner. Yes?

2. The graph of y = -tan x is very similar to
y = cot x on the xy-plane. By "similar" I mean they travel from left to right in a curve-like manner. Yes?

3. Why does the tangent function have asymptotes?

4. Same question as 3 above concerning the cotangent function. Why?

5. An asymptote is a line that functions do not cross or touch. If this is true, why is this the case?

Thanks

 

[MathLover1]

1. correct
2. correct
3. recall that tan(x) =sin(x) /cos(x) and cos(pi/2) =0
that is why the graph of y=tan(x) has discontinuities at x=π/2
4 same reason as in 3, just sin(pi) =0 and the graph of y=cot(x) has discontinuities at x=π
5 that is true
why is this the case? because of discontinuities mentioned above (asymptote is at the points of discontinuities)

 

[nycmathguy]

1. correct
2. correct
3. recall that tan(x) =sin(x) /cos(x) and cos(pi/2) =0
that is why the graph of y=tan(x) has discontinuities at x=π/2
4 same reason as in 3, just sin(pi) =0 and the graph of y=cot(x) has discontinuities at x=π
5 that is true
why is this the case? because of discontinuities mentioned above (asymptote is at the points of discontinuities)

Thank you. Great reply.