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Expert: - 3/13/2008


Question
I originally thought that by replacing the unit circle's x values with radians/degrees and keeping the y values the same was all that there was to sin and cos graphs.  Is that right?

Thanks!!

Answer

Sines
Questioner:   Bennett
Category:  Advanced Math
Private:  No
 
Subject:  How sin and cos graphs are formed.
Question:  I originally thought that by replacing the unit circle's x values with radians/degrees and keeping the y values the same was all that there was to sin and cos graphs.  Is that right?

Thanks!!
...............................
Hi, Bennett,

Well, there is just a bit more.  Do you have a spare month or so to work on it?

Here is what I recommend: If you want to graph:

y = A sin (bx)

A is called the Amplitude.
b is called the Frequency.
P (you don't see it yet) is the Period

and you get the period by solving the equation  bP = 2 pi.
(Yes, it's always 2 pi.)

You get the basic shape of the graph by looking at a few.  I have attached a picture of some sine curves.  Here is what you do, applied to graphing

y = 3 sin (2x), which is not one of them in the picture.

Compute the period:  2P = 2pi,  P = pi. (radians, of course.  Degrees are for children.)



Draw some nice axes like this:

 +
 |
 |
 |
 |
 +----------------------------------------
 |
 |
 |
 |
 +

Mark the period, in this case, pi, somewhere at the right.

 +
 |
 |
 |
 |
 +-------------------------------------+---
 |                                     pi  
 |
 |
 |
 +

Mark HALF the period, halfway (of course.)

 +
 |
 |
 |
 |
 +-------------------+-----------------+---
 |                  pi/2               pi  
 |
 |
 |
 +

Mark HALF OF HALF the period, which is a quadrant.

 +
 |
 |
 |
 |
 +---------+---------+-------------------+---
 |       pi/4       pi/2                 pi  
 |
 |
 |
 +

Mark THREE TIMES the quadrant in the right spot.

 +
 |
 |
 |
 |
 +---------+---------+---------+---------+---
 |       pi/4       pi/2     3pi/4       pi  
 |
 |
 |
 +
Label the y-axis with the amplitude:

3+
 |
 |
 |
 |
 +---------+---------+---------+---------+---
 |       pi/4       pi/2     3pi/4       pi  
 |
 |
 |
-3+

From your knowledge of the SHAPE of a sine curve, put in the key points -- the zeroes and the max and min:

3+         O
 |
 |
 |
 |
 O---------+---------O---------+---------O---
 |       pi/4       pi/2     3pi/4       pi  
 |
 |
 |
-3+                             O

Last, draw a graceful curve of the proper shape.

There is a bit more, but this will get you started.  Only 29 more days to go and you have it all.

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

Experience

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

Education/Credentials
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