Why Learn Trigonometry?

We have spoken a great deal about visualizing functions and trigonometry concepts, but let's take a step back for a moment and see the bigger picture.

Trigonometry did not click for a long time for me.  To this day, I am still relearning the simplest ideas and seeing them in new ways.  The unit circle is nice to learn, as it gives us a great foundation and the number 1 is easy to work with, but there is another fascinating parallel we can make.

The height of the human body is roughly equal to our arm span.  If we draw a square and a circle around a human figure, we can see two very distinct relationships, and this is where ancient mathematicians discovered a great deal of their insight and wisdom.

Vitruvian Man
Retrieved from https://en.wikipedia.org/wiki/Vitruvian_Man#/media/File:Da_Vinci_Vitruve_Luc_Viatour.jpg

If we stand with legs together and arms straight out, we form a square, and if we angle our legs and arms, we maintain the same relationship, but since we are moving a constant radius, we are moving our limbs along a circle.

If you have trouble memorizing the trig functions in the way we discussed earlier, perhaps we can visualize them another way.

Pretend we are in the middle of a dome.

Now pretend someone has built a wall at the edge of our dome.  It would appear like this:

Wall Example
Retrived from http://betterexplained.com/wp-content/uploads/trig/trig-wall.png

In this example, we can visualize tangent and secant in a new way.  Suppose the wall is blocking our vision.  To climb the tangent wall (green), we'll build a secant ladder (red).  Now you can SEE, CAN'T you?  Get it?  Secant?  Tough crowd...

I didn't come up with it, but thank Kalid Azad at http://betterexplained.com/articles/intuitive-trigonometry/.  I highly recommend reading this site for a more thorough explanation.

Now that we have seen tangent and secant, cosecant and cotangent are not far off.  Instead of a wall, let's pretend a ceiling was built instead.  Imagine that it extends outward beyond the edge of the dome.

Ceiling Example
Retrieved from http://betterexplained.com/wp-content/uploads/trig/trig-ceiling.png

In this example, the height of the ceiling is the radius of the dome (green).  This could easily have been built at the origin, but placing it outward better illustrates the red ramp, which is the cosecant.  If we have a given angle, the cotangent is the length of the base of the ramp, and the cosecant is the length of the ramp itself, rising toward the ceiling.

When we put both of these drawings together, we get a brilliant illustration of all the trig functions relating to one another, all forming similar triangles which can easily be solved with simple ratios.

Combined Graph
Retrieved from http://betterexplained.com/wp-content/uploads/trig/trig-overall.png

From here, we can actually re-derive the trig identities we learned last time in a much simpler way than memorizing.  Take the blue segment, the green segment, and the red segment connecting the two, and we form all three of the Pythagorean Identities.

Consider the original Pythagorean Theorum:
a^2 + b^2 = c^2

cos^2(x) + sin^2(x) = 1

1 + tan^2(x) = sec^2(x)

cot^2(x) + 1 = csc^2(x)

It's pretty amazing when we put them all together in this way.  Trig functions can be derived in so many ways because triangles and circles are so closely related.

Sources: 

Kalid, Azad. How To Learn Trigonometry Intuitively. Retrieved November 30, 2015, from http://betterexplained.com/articles/intuitive-trigonometry

Graphing Trig Functions

Now that we have been introduced to the reciprocal trig functions, secant, cosecant, and cotangent, let's consider their graphs.

Recall with sine and cosine, we saw their relationship was very closely related, especially over the period of 2π when studying the unit circle.
I always remembered sine as a sideways "S", since it starts with S.

Similarly, for cosine, I always thought of it as a "Crow" or a bird, since it looks like a bird flying in the distance and it conveniently starts with a "C".

In any case, sine and cosine are simple functions to draw.  If we remember our unit circle values, sine is 0 at 0, π, and 2π, and is 1 at π/2 and 3π/2.  Draw a dot for each of those points like you would for any other function, and just do your best to connect the dots with a curve.  The cosine graph is the same, but where sine is 1, cos is 0, and where sine is 0, cos is 1.



What about secant and cosecant?  If we never knew what sine and cosine looked like, we would never be able to make sense of them, but let's see what they look like when considering their reciprocals.

Though they look like U's, they are actually distinct parabolas heading toward infinity.

Cosine = Adjacent / Hypotenuse.   When the Adjacent side is 0 on the unit circle, the angle is 90 degrees, and the hypotenuse is 1.  We can divide 0 by 1, and it equals 0.  What happens when you flip them?  We get 1 divided by 0, which is impossible.  Let's get as close to 0 as we can within reason though, and divide 1 by .000000001 - a very very small number.  We get 1000000000, a very very large number.  The closer and closer we get to 0, the larger and larger the number gets.  So we get a vertical asymptote everywhere that cosine = 0.

It is for this reason that we can simply draw parabolas on each bend of the sine and cosine curves.  I apologize for carrying on so long, the important thing to take from this is to understand why sine and cosecant are related, and cosine and secant are related.

To draw the curves (Let's graph sec(x)): 

1.)  Draw the reciprocal function (red).  Secant relates to cosine, so pencil in the graph of cosine along 0 to 2π.  We choose 0 and 2π because this is the period of the graph.

2.)  Pencil in the vertical asymptotes.  Wherever the graph of cosine is 0 on the y-axis, the secant will approach infinity.  These are the black lines in our sample graph.

3.)  Draw parabolas reflecting out of the graphs curves (shown in green).  These are the periodic minimums and maximums of the graph, also known as the magnitude.  The magnitude for trig functions on the unit circle are simply 1.

The accuracy of hand-drawn graphs is not particularly important, as long as you understand the values where the graph hits 0 and 1, that is what is important - most notably at 0, π/2, π, 3π/2, and 2π.


Graphs created with Desmos Graphic Calculator.
https://www.desmos.com/calculator

Secant, Cosecant, and Cotangent : Simplifying Trig Functions

As we explore trig identities and their graphs further, it is important to emphasize how each of them relate to one another.  When we look at any given right triangle, we know that sine = the opposite side (from a chosen angle) over the hypotenuse, which is the longest side.  Cosine = adjacent over hypotenuse, and tangent = opposite over adjacent.

SOH CAH TOA retrived from:
http://mathworld.wolfram.com/SOHCAHTOA.html

SOH - CAH - TOA is an extremely useful mnemonic to remember these relationships, and it is likely you have already been introduced to this simple reminder.  However, if we flip those fractions, we create three more unique functions.

Sine (sin) -->  Cosecant (csc)  :  sin=O/H  --> csc=H/O

Cosine (cos) --> Secant (sec)  :  cos=A/H --> sec=H/A

Tangent (tan)  -->  Cotangent (cot)  :  tan=O/A --> cot=A/O

It always helped me to remember if we flip the fraction, you add "co" to the beginning, or take it away.  Sine becomes CO-secant.  Cosine already has "co", so we take it away, and it becomes secant. Tangent turns to CO-tangent.

These are derived by using laws of multiplying and dividing fractions.  Using trig identities, we can easily cancel functions out and simply many hideous and scary looking formulas.  By writing all of these functions in terms of sine and cosine, we can work with something much easier and more familiar to us.  We'll prove them in class, but for quick reference:

tan = sin/cos, thus

cot = cos/sin

Also, remember that as these are reciprocals, remember that these functions can be written as 1 over their reciprocal.  

For example:  
1 / sin = csc
1 / cos = sec
1 / tan = cot
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3 of the most important trig identities are based off these laws, and thankfully there are more dorky ways to remember them.  These are the Pythagorean Identities, and are derived from playing around with the simple Pythagorean Theorum we all know;  a^2 + b^2 = c^2

Retrieved from
http://www.rachel.worldpossible.org/modules/olpc/wikislice-en/files/articles/Trigonometry.htm

sin^2 + cos^2 = 1, we simply have to remember, and it will become the most familiar identity.  The other two are not as simple though.  Here is how I always remembered them:

1+tan = sec -->  1 TAN man has a SECret.

1+cot = csc -->  1 man on a COT Cannot Sleep Comfortably.

It's a bit of a stretch, and the first time I heard it I hated it, but it was so ridiculous that I remembered it for the rest of my life.  All that is important is the 1+TAN=SEC, and 1+COT=CSC, and to remember that these quantities are squared.  If this doesn't work for you, feel free to memorize it any way that works for you.
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Preview and Practice: (http://home.windstream.net/okrebs/page84.html)

This site has a few brilliant examples of how we can rewrite in terms of sine and cosine, to begin to get a handle on how these functions relate to one another.  It's hard to see why this skill is useful right now, but trig identities become a very powerful tool in higher mathematics - especially if you plan on continuing to study through calculus.