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Question 1083716: Use a reference rectangle and the rule of fourths to draw an accurate sketch of the following function through at least one full cycle.
y = cos(2t)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the general form of the cosine function is y = a * cosine (b * (x-c)) + d
a is the amplitude whose default value is 1.
b is the frequency whose default value is 1.
c is the horizontal shift whose default value is 0.
d is the vertical shift whose default value is 0.
when the values are default, they are not shown.
therefore, y = cos(x) has a default value of a = 1, b = 1, c = 0, and d = 0.
an amplitude of 1 means the cosine function travels 1 unit above the center line of the function and 1 unit below the center line of the function.
a frequency of 1 means the interval of the cosine function is 2pi.
that means the function makes one full cyles in an interval of 2pi along the x-axis.
a horizontal shift of 0 means the cosine function starts a full cycle when x = 0.
a vertical shift of 0 means the cosine function center line is at y = 0.
the following graph shows you the function of y = cos(x).
you can break the cosine function interval into quarters.
the normal interval is from 0 to 2pi.
break that up into quarters and you get quarterly intervals of:
0 to pi/2
pi/2 to pi
pi to 3pi/2
3pi/2 to 2pi
for the cosine function, these quarterly intervals are useful because:
when x = 0, cos(y) = 1
when x = pi/2, cos(y) = 0
when x = pi, cos(y) = -1
when x = 3pi/2, cos(y) = 0
when x = 2pi, cos(y) = 1
when you draw the graph, you will see that the cosine function starts at 1 and goes through a complete cycle where it lands back at 1.
the graph of y = cos(x) is shown below:
the general form of the cosine function is, once again, y = a * cos(b * (x-c)) + d
when b = 2, the frequency of the cosine function within the normal interval is doubled.
this means you get two full cycles within the normal interval of 2pi.
the interval of one full cycle of the cosine function is therefore cut in half.
the formula for interval is:
interval = 2pi /frequency
when frequency is 1 (default), the interval is 2pi / 1 = 2pi.
when frequency is 2, the interval for one full cycle becomes 2pi / 2 = pi.
if you break the interval of 0 to pi into quarters, you will get:
0 to pi/4
pi/4 to pi/2
pi/2 to 3pi/4
3pi/4 to pi
the value of the cosine function will now be:
y = 1 when x = 0
y = 0 when x = pi/4
y = -1 when x = pi/2
y = 0 when x = 3pi/4
y = 1 when x = pi
the graph of y = cos(2x) is shown below:
having an interval length of half the normal interval length means you can get two full cycles of the cosine function in the same interval that you normally get one full cycle of the cosine function.
this is illustrated in the following two graphs.
you can see that, in the normal full cycle of 0 to 2pi for the cosine function, cos(x) goes through one full cycle and cos(2x) goes through 2 full cycles.
where does the rectangle fit in?
here's a view where the full cycle is bounded by shaded regions on the left and right and on top and bottom.
you can see that the cosine function is within what looks like a rectangle.
i believe that might be what they're talking about.
if you look at this graph, the unshaded region is where one full 2 full cycles of the cosine function reside.
the function is bounded by a box that looks like it is in the shape of a rectangle.
here's a reference that i found regarding the graphing of the sine and cosine function that you might find helpful.
http://www.purplemath.com/modules/grphtrig.htm
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