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### Graphing Sine and Cosine

Date: 07/07/2000 at 13:56:36
Subject: Graphing sine and cosine

I just don't understand how my teacher gets the points that he uses on
the graph. I have tried to do the same thing he does over and over

Date: 07/07/2000 at 14:34:56
From: Doctor Jaffee
Subject: Re: Graphing sine and cosine

I assume that you are talking about sketching the graph of a sinusoid;
that is, an equation in one of the following forms:

1. y = C + AsinB(x - D)
2. y = C + AcosB(x - D)

Here is how I would draw the graph.

First, draw a dotted horizontal line C units above (or below if C is
negative) the x-axis. This dotted line is the transverse axis. Half of
the graph will be above the dotted line and half will be below.

If D is positive, go D units to the right of the origin and, if you
are drawing a type 1 equation (the one with the sine), go up to the
transverse axis and put a dot. This will be the first point of the
cycle. If D is negative, go to the left.

If your equation is type 2, go D units to the left or right just as
I suggested for type 1. Then if A is positive, put the dot A units
above the transverse axis. If A is negative, the point will be below
the transverse axis.

Now, if you are doing a type 1 graph and x is in degrees, then the end
of the cycle will be on the transverse axis, 360/B degrees to the
right of the first point of the cycle. Halfway between these two
points on the transverse axis will be the midway point of the cycle.
If the equation is in radians, then the last point will be 2*pi/B
units to the right of the first point.

Halfway between the first point of the cycle and the midway point of
the cycle, draw a point A units above (or below, if A is negative) the
transverse axis.

Halfway between the midway point and the end of the cycle, draw a
point A units below (or above, if A is negative) the transverse axis.

By now you should have 5 points. If you draw a smooth curve connecting
them, you will have one cycle of the sinusoid.

If you are graphing a type 2 sinusoid, then the first and last points
of the cycle will be A units above the transverse axis and the midway
point will be A units below the axis. Of course, if A is negative,
just the opposite will happen. The points 1/4 and 3/4 of the way
through the cycle will be on the transverse axis.

I hope this explanation helps. Try a few examples and see what
happens. If you are having difficulties or if some of my explanation
was unclear, write back and I'll try to help you some more.

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
High School Equations, Graphs, Translations
High School Trigonometry

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