__The premise__:
Let **f(x) = ****a sin(bx + c)** and **g(x) = ****a cos(bx + c)**.

For selected values of **a**, **b**, and **c**, graph and explore

i. the sum of
** f** and

ii. the product
of ** f** and

iii. the quotient
of ** f** and

iv. the composition
of ** f** with

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To begin this exploration, let **a**
= **b** = 1 and **c**
= p/4. In the image
below, the purple graph is f(x) = sin(x + p/4); the red graph is g(x) = cos(x + p/4); the blue graph
is ** h** =

** h**
appears to be a cosine graph with amplitude larger than

**Vary** **a**: Now, keeping
**b** and **c** constant, vary **a**. In the graphs below,
**a** varies from 0.5 (green graph) to 1 (purple) to
2 (red) to 3 (blue.)

As would be expected, varying **a**
changes the amplitude of the resulting cosine graph. In fact,
the amplitude of these sums of ** f** and

**Vary** **b**: Next, keep
**a** = 1, **c**
= p/4, and vary
**b**. In the graphs below, **b** varies from 0.5
(green graph) to 1 (purple) to 2 (red) to 3 (blue.)

As would be expected, varying **b**
varies the period of the resulting cosine graph. All these graphs
are the result of a horizontal shrinking or stretching. So, these
graphs of the sums of ** f** and

**Vary** **c**: Finally, keep
**a** = **b**
= 1 and let **c** vary. In the
graphs below, **c** varies from
p/2 (green graph)
to 3p/8 (blue)
to p/4 (purple)
to p/8 (red).

It is no surprise that varying **c**
affects the phase (horizontal) shift of the graph, but it's not
so immediately obvious how the graph is shifting. One way to think
about it is to determine what x-value will now yield p/4 as an argument for
sine or cosine, since that x-value will then produce the maximum
on the graph of the sum of ** f** and

Now, put it all together! Below are graphs with
varying values for **a**, **b**, and **c**. The equations are listed to the right.

So, if **f(x) = ****a sin(bx + c)**
and **g(x) = ****a cos(bx + c)**, then the sum of
** f** and

To see an algebraic proof, click **here**.

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To begin this exploration, again let **a**
= **b** = 1 and **c**
= p/4. In the image
below, the purple graph is f(x) = sin(x + p/4); the red graph is g(x) = cos(x + p/4); the blue graph
is ** h** =

The result is a cosine graph with amplitude
1/2, since again the maximum point of ** h** will occur
when the y-values of the sine and cosine graphs are equal, i.e.,
when sine and cosine both yield sqrt(2)/2. [sqrt(2)/2 * sqrt(2)/2 = 1/2.] The
period of the graph is p, which means an equivalent equation for this product
of

**Vary** **a**: Now, keeping
**b** and **c** constant, vary **a**. In the graphs below,
**a** varies from 0.5 (green graph) to 1 (purple) to
2 (red) to 3 (blue.)

Again, **a** affects the amplitude of the graph because it stretches
or shrinks ** f** and

**Vary** **b**: Next, with
**a** = 1 and **c** = p/4, vary **b**. In the graphs below, **b** varies from 0.25
(blue graph) to 0.5 (green) to 1 (purple) to 2 (red.)

It is no surprise that varying **b**
varies the period of the graph, since it produces a horizontal
stretch or shrink. When **b** = 1, the period of ** h** is p; when

**Vary** **c**: Finally, keep
**a** = **b**
= 1 and let **c** vary. In the
graphs below, **c** varies from
p/2 (blue graph)
to 3p/8 (green)
to p/4 (purple)
to p/8 (red).

Again, the phase shift of the graph is affected.
To analyze the situation, consider that we are still interested
in what value of x produces p/4 as an argument for sine and cosine. That is,
the argument of sine or cosine will be p/4 when x = p/4 - **c**. So, these graphs are equvalent to **h(x) = 1/2
cos(2(x - (****p/4
- c)))**. While this way of writing the equation is somewhat
cumbersome, it allows for the identification of the various effects
of changing **a**, **b**, and **c**.

Now...view some examples below, where **a**,
**b**, and **c** vary.

I conjecture that if **f(x) = ****a sin(bx + c)** and **g(x) = ****a cos(bx + c)**,
then the product of ** f** and

To see an algebraic proof, click **here**.

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The exploration of this combination of ** f** and

Notice that when we take the quotient of ** f** and

In the purple graph, In the blue graph, |

In the red graph, In the green graph, |

As you can see, **b** still affects the
period of the tangent graph, and these graphs all have periods
of p/**b**.
The phase shift of the graphs is -c/**b**.

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Now, this part of the exploration is where the *really*
cool stuff begins. Again let **a**
= **b** = 1 and **c**
= p/4. In the image
below, the purple graph is f(x) = sin(x + p/4); the red graph is g(x) = cos(x + p/4); the blue graph
is h(x) = f(g(x)).

Obviously, the composition is no simple cosine graph!

**Vary** **a**: Keeping **b**
and **c** constant, vary **a**. See
a movie of how ** h** changes with varying values
of

Wow! On the graph, n =

aranges from 1 to 8. Notice that the maximum and minimum values of the graph when n = 8 appear to be, respectively 8 and -8. Obviously, changingastill affects the vertical stretch of the graph, but it also appears to affect the number and direction of the "dips" in the graph, and the horizontal compression of the graph within the viewing frame. (Of course, vertical stretching can give the impression of horizontal compression.) On the same window, for smaller values ofathe graph appears more "spread out," while for larger values ofathe graph appears more compressed. Also, asaincreases, there appear to be more dips--i.e., more maximum and minimum points.

**Vary** **b**: Next, with
**a** = 1 and **c** = p/4, vary **b**. See a movie of how
** h** changes with varying values of

With the sliding values of n =

b(from 1 to 4), the horizontal compression asbincreases can be seen very clearly. Asbincreases, the graph becomes very compressed with the dips appearing more numerous and pronounced. Asbdecreases, the graph appears to smooth and spread out. Interestingly, atb= 1, the maximum appears to be about 1 and the minimum about -0.5. Asbincreases, the maximum value does not appear to change but the minimum value drops to approximately -1, indicating that changingbaffects more than just horizontal compression.

**Vary** **c**: Finally, keep
**a** = **b**
= 1 and let **c** vary. See a movie of how
** h** changes with varying values of

The sliding values of n =

c(from 0 to 15) clearly display why sines and cosines are referred to as wave functions! Varyingcappears to affect both the horizontal and vertical translation of the graph, but it also affects the shape. Ascincreases, the graph appears to translate to the left, (but also up and down); whencdecreases, the graph appears to translate to the right, as well as up and down. Whenc= 0, and periodically thereafter (aboutc= 3, 6, 9.45, 12.45), the graph resembles a simple sine wave with an amplitude nearly equal to 1. In between those values ofc, (at aboutc= 1.5, 4.5, 8, 11), the graph again appears to resemble a simple sine wave with an amplitude of approximately 1/4, translated up or down off the x-axis. Socdefinitely affects the shape of the graph, not just the shift.

Now...view some examples below where **a**,
**b**, and **c** vary.

In the purple graph, In the red graph, |

In the blue graph, In the green graph, |

I can only conjecture, at this point, that
the composition of ** f** and

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