The best thing about sinusoidal functions is that you don't have to plug in random values or make a table. There's only three key parts:

Here's the parent function for a sinusoidal graph:

#color(blue)(f(x)=asin(wx) color(red)((- phi) + k)# Ignore the part in red

First, you need to find the period, which is always #(2pi)/w# for #sin(x), cos(x), csc(x), and sec(x)# functions. That #w# in the formula is always the term next to the #x#. So, let's find our period:

#(2pi)/w = (2pi)/3#. #color(blue)("Per. T" = (2pi)/3)#

Next, we have the amplitude, which is #a#, and generally in front of the trigonometric term, and what the y-coordinates will be every other point. The amplitude can be regarded as the max and min of the graph, as seen above.

So, now we have our amplitude. #color(blue)("Amp."=1)#

When you make a sinusoidal graph, the period will be four x-coordinates to the right and left.

Start with the fourth point, as seen above, which is your period, #color(blue)((2pi)/3)#

Then go to the second point, which is half the period: #color(blue)(((2pi)/3)/2 = pi/3)#

Then go to the first point, which is one fourth the period (or half the second point: #color(blue)((pi/3)/2 = pi/6)#

Now we have our five key points in terms of #color(blue)(pi/6):#

#color(blue)((0,0) (pi/6, 1) (pi/3, 0) (pi/2, -1) ((2pi)/3, 0))#

This is the same as:

#color(blue)((0,0) (pi/6, 1) ((2pi)/6, 0) ((3pi)/6, -1) ((4pi)/6, 0))#

Just notice that the top values are simplified to what the graph shows.

Another important thing to remember is that #Sin(x)# graphs start at the origin and progress upward, unless the amplitude is negative, then they would progress downward. #Cos(x)# graphs start at #(0, "Amplitude")# and move downward, unless the amplitude is negative, then it would start at #(0, "-Amplitude")# and move upward.