Sine and Cosine: How do they interact?

Hello again, class!  I'd like for us to take a closer look at the graphs of sine and cosine functions. What can we learn from studying their relationship? We know that they form a circle somehow; and as one increases one decreases, but why is that? Hopefully we make life a little easier for you in the coming lessons by visualizing them.

When we study the unit circle, we see how sine and cosine are directly related in terms of x-y coordinates.  This is the rectangular coordinate system that we are used to, where we draw a grid.

Unit Circle
http://mathmistakes.info/facts/TrigFacts/learn/images/ucdefp.gif

When we start at 0, the value of cosine is 1, and sine is 0.  As we increase the angle, we climb the circle upwards, or increase the sine.  We notice from here that the cosine begins to shrink until we hit 90 degrees, where sine is now 1, and cosine is now 0.  As we continue along the circle, sine shrinks and cosine starts going into negative values until we hit 180 degrees.  Continuing along, sine and cosine are both negative and finally, we enter the 4th quadrant where the circle closes up.

Sine and Cosine (+ and -)
http://aventalearning.com/courses/ALG2x-HS-A09/a/unit05/resources/images/A2_5_3_10_unitCircle.jpg

This relationship might be hard to conceptualize at first, but here is a brilliant image of a machine that roughly illustrates sine and cosine's relationship in the Unit Circle.

Sine and Cosine.
  http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5

Notice how this relationship continues on infinitely.  The domain, or set of inputs that defines a function, is defined for all values of theta.  That means we could calculate the sine and cosine values for -360000000 degrees, and 360000000 degrees, or any infinitely large value you could imagine.

Practice: Try putting a random number into your calculator and pressing the sin key.  For example, let's try 60.

sin(60)=0.8660254...

Let's add 360 degrees to that, or a complete turn around the circle.  360+60=420

sin(420)=0.8660254...

Let's add 360x100 to that.  360x100 = 36000+420 = 36420.

sin(36420)=0.8660254...

We get the same result!  You calculator is actually solving for unit circle values for these functions. Consider the following image as well:

Sine and Cosine
http://static4.businessinsider.com/image/51910f38eab8ea4c31000002/0tlnknd%20-%20imgur.gif

We can see, as the circle is traced, it's "shadow" or "projection" on the yz axis (drawn in red) is actually the sine function, and its projection on the xz axis (drawn in blue) is the cosine function! When we visualize it this way, we see how these infinitely occuring functions occur - it is because we can trace infinitely many circles in space.

Doesn't it look like a spring?  It's not a far stretch to assume that trig functions can be applied to physics and mechanics.  That is only the beginning - the usefulness of these functions goes on forever, just like the functions themselves.

Sound acts in waves - light acts in waves; there is so much about the world we can discover through these ideas!  Any circular pattern we observe in the world can likely be modeled and adapted to trigonometry; even those in nature.  Concepts like shadows, hours of sunlight, and even wildlife population can be studied with trigonometry.