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

1 Answer
Mar 15, 2018

Answer:

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