trigonometry
posted by Math .
I need help finding the key points for the graph y=1/3 cos x and y=sin 2piex/3
I just don't understand how to graph this and find keypoints.

y = 1/3 cos x.
You know that cos x = 0 when x = pi/2, 3pi/2, etc. All odd multiples of pi/2
It is 1 for all even multiples of pi
It is 1 for all odd multiples of pi.
But you have y = 1/3 cos x, so it is 1/3 or 1/3 at multiples of pi
Easy, right?
Now, for sin 2pi/3 x, things do get a bit trickier.
You know that for sin x, graph crosses xaxis at all multiples of pi
y=1 for all 4k+1 * pi/2
y=1 for all 4k1 * pi/2
y = sin x will look the same, but the max/min will be reversed.
But, we have sin 2pi/3 x
So, graph will cross xaxis whenever 2pi/3 x is a multiple of pi
That is, 2pi/3 x = k*pi, so x = k*pi * 3/2pi = 3k/2 for all integer values of k
y = 1 when 2pi/3 x = (4k+1)pi
x = 4k+1 pi * 3/2pi = 3/2(4k+1) = 6k + 3/2
y = 1 when 2pi/3 x = 4k1 pi, so x = 6k  3/2
To do this kind of scaling/translating, it's usually easy in the beginning to work with what you know (sin x), and then step by step transform from the new variable.