Question

How do you graph two cycles of `y=2tan(3theta)`?

Answer 1

Try sketching with references to certain properties of the graph, noticeably intersects, monotonies, and asymptotes.
graph{tan(3x) [-2.23, 2.23, -1.162, 1.162]}

Explanation:

Note that all `theta` has been replaced with `x` in the following explanation.

1. Intersections
Evaluate the function at `x=0` to find the `y` -intersect:
`tan0=0` thus the function intersect the `y` -axis at `(0,0)` and therefore passes through the origin.

We can find `x` -intersects, or "zeros," of the function by setting its value to zero and solving for `x`.
Let `tan(3*x)=y=0`

This explanation shows how to solve the equation by considering the composite nature of the function: it consists of two parts, an inner function `f(x)=3*x`, and an outer function `g(u)=tan(u)`. So the original function, `tan(3*x)`, is equivalent to `g(f(x))`. Thus values of `u=f(x)` at zeros of this functions shall ensure that `g(u)`, the outer function gives a value of zero.

`tan(u)=0`,
`u=0+k*pi`, where `k` is an integer (`k in ZZ`). Here `u` has more than one possible value since the function `tan u`, is cyclical with a period of `pi`.

Substituting `u` with an expression about `x` gives
`3*x=k*pi`
`x=k/3*pi`

Thus coordinates of `x` -intercepts of this function shall fit into the general expression
`(k/3*pi,0)`

Taking `k=-1`, `k=0`, and `k=1` gives `x` -intersects
`(-k/3*pi,0)`, `(0,0)`, and `(k/3*pi,0)`.

2. Monotonies
The tangent function always increases as the angle grows, as seen from a unit circle. Therefore the graph of `tan x` should slope upwards and extend to the upper-right corner of the Cartesian plane. This observation is also the case for a composite tangent function like `tan u` as long as the inner function `u=f(x)` is increasing.

3.Asymptotes
The tangent function is not defined at the sum of `pi/2` any integer multiple of `pi` and shows asymptotic behavior at each of these values of `x`. That is
`x=pi/2+k*pi` where `k` is an integer.

For the composite tangent function here, the general expression for all the asymptotes would be
`3*x=u=pi/2+k*pi` and
`x=(1/6+k/3)*pi`

Evaluating at the expression at `k=-1`, `k=0` and `k=1` gives
`x=-1/6*pi`, `x=1/6*pi`, and `x=1/2*pi`.

Now plot all three of these features on the graph, and the curve you sketch should:
a. Slopes upwards;
b. Passes through all of the intersections, and
c. Approaches, but never touches each of the asymptotes.

See also:
https://www.mathsisfun.com/geometry/unit-circle.html
Value of the tangent function on a unit circle