MY FAVORITE TAKEAWAY

We come to the end of the semester, and it seems like the time to share the pinnacle of the knowledge I will take away from this experience: THE TRIGONOMETRIC CIRCLE!!!  I have always been a visual learner, and to see the interconnectedness of all the principal trig functions as they relate to a unit circle was, well, like oxygen to a flame.  Suddenly it all made sense, and all I could think was, "Man, if I had been shown this in high school, I wouldn't be struggling so much with this now!"  And now I would like to share it with the world!

 

Okay, enough hyperbole.  Now let's explain how it works.  But first, let's start with the basics, the relationship of sine and cosine.  Take a look at this graphic:

 

This shows how sine and cosine cycle between 1 and -1 as we travel around the unit circle.  As the point on the perimeter travels towards the x-axis, sine approaches 0 and cosine approaches 1 or -1.  As the point approaches the y-axis, sine approaches 1 or -1 and cosine approaches 0.  It is from this basic relationship that all the other trig relationships are derived.

First, there's tangent, which is sine/cosine.  So, as sine approaches 1 or -1 and cosine approaches 0, tangent approaches positive or negative infinity, but it is undefined when cosine equals 0.  Conversely, as sine approaches 0 and cosine approaches 1 or -1, tangent approaches a value of 0. 

Second, there's cotangent, which is 1/tangent, or cosine/sine.  This means when sine is getting smaller and smaller, cotangent approaches positive or negative infinity, and it approaches 0 as cosine approaches 0.

Thirdly, there is cosecant, which is 1/sine.  We can see from the values of sine that cosecant can never be smaller than 1 or -1, and it stretches towards infinity as sine closes in on that zero value.

Fourthly, there is secant, which is 1/cosine.  Again, secant is never less than 1 or -1, and it, too, goes on to infinity as cosine approaches 0.

So, what do all of these six (!) functions look like on the unit circle?  See below:



Play around with the circle for a spell, especially in the first quadrant.  What do you notice?  As expected from our first graphic, sine and cosine oscillate between 1 and 0 and -1, depending on which axis we are approaching.  But it is the other four derived trig functions that are really interesting to observe here.  As the pink point moves closer and closer to the x-axis, sine approaches 0 and cosine approaches 1.  Take a look at the green tangent line; it approaches a length of zero as well, which is to be expected when we recall that tangent is sine/cosine.  Cotangent, aka tangent's inverse, heads towards infinite length, a likewise unsurprising result.  Secant, derived from cosine, heads towards its minimum value of 1, and cosecant, derived from sine, heads towards an infinite length.
Now, move the pink point to where the angle centered at the origin is 45 degrees, or as close as the graphic will allow (I have been unable to get closer than 45.04 degrees).  Notice the marvelous square symmetry of it all.  Sine is equal to cosine, secant is equal to cosecant, and tangent is equal to cotangent.  Again, all of this makes perfect sense when we recall how each function is derived, but now we can see that similarity in visual form. 
Finally, start moving the pink point towards (0,1).  Notice sine approaches 1, cosine approaches 0, tangent moves towards infinite length, cotangent towards zero, secant is likewise approaching infinity, and cosecant is heading towards a value of 1. 
The truly exciting aspect of this graphic for me is how simple it is to derive other trig identities by using a little Pythagorean algebra.  For example, sine and cosine are always two legs of an inscribed right triangle with a length of 1, so we can derive that sin²(a) + cos²(a) = 1.  We can also observe that tangent and the radius of the circle form the legs of a right triangle with secant as the hypotenuse, so we can now derive that sec²(a) = tan²(a) + 1.  The same process gives us csc²(a) = cot²(a) + 1.  Even if we forget the identities themselves, we can very easily figure them out if we remember visually exactly what is going on with the unit circle. 

So that's it.  My most fulfilling learning experience of the semester.  If and when I get around to teaching trigonometry, the unit circle seen here will definitely be my go-to learning device for helping students grasp the trig functions.  Hopefully I can save all my students the years of anguish and confusion I went through. 
Thanks for reading; it's been fun.  Cheers!