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.