Math, The Way it Should Be (Unit 2.4: Trigonometry)

It's been a while since the last math post but that's an important one for this lesson, so if you missed it go ahead back and check it out.  As usual, the first post in the whole series is here and the first post in this unit is here.  This post is going to be a very brief overview of trigonometry.  This is usually a subject covered over the course of a semester so it would be rather silly to try to cover the whole thing in one post so you'll get the highlights.  There's going to be a lot in this lesson.  Ready?

First off, trigonometry is basically the study of triangles.  Remember those?  There is a basic refresher here.  It also talks a bit about radians which will be really important now and are covered in the first post of the unit (link in the first paragraph).  Now there are some special functions that come about in trigonometry and they're aptly named, the trigonometric functions.  The first three are fairly basic: the sine, the cosine, and the tangent functions. They are usually seen like this: \(sin(x)\), \(cos(x)\), and \(tan(x)\).  What do they do?  Well they map the angles to ratios.  Let's look at the triangle we had last lesson:

In this case, let's say that the letters represent the length of the sides and they can also serve as the side's name.  Let's also label the angles (the names will also represent the measure of that angle in radians):

Now we're in business.  When we start with trigonometry, we usually start with a right triangle.  Therefore, let's assume that \(Y = \frac{\pi}{2}\).  This angle won't factor heavily into the discussion but this is an important step to take.  Now, let's put these definitions to work defining the trigonometric functions.  The sine function takes a given angle and spits out the ratio of the opposite side to the hypotenuse of a hypothetical triangle with that angle.  For example, \(sin(Z) = \frac{a}{c}\).  In the same way, \(sin(X) = \frac{b}{c}\).

The cosine function is equally easy.  It takes the angle and spits out the ratio of the adjacent side and the hypotenuse.  In this case, \(cos(X) = \frac{a}{c}\) and \(cos(Z) = \frac{b}{c}\).  The observant reader will notice that \(cos(X) = sin(Z)\) and this will always be true for any two positive angles that add up to \(\frac{\pi}{2}\).  You'll also notice that \(sin(X) = cos(Z)\) and the exact same logic applies.  This is not necessarily true for all angles as the functions are extended to all real and complex angles!

The last of the basic trigonometric functions is the tangent function.  It returns the ratio of the two legs of the triangle (in other words, the two sides that aren't the hypotenuse).  \(tan(X) = \frac{b}{a}\) and \(tan(Z) = \frac{a}{b}\).  And there you have it, the first three trigonometric functions.  They are the ones that are seen most often.

What about the others?  There are many many many.  The next three return the inverse of their basic counterpart:
secant:
\(sec(x) = \frac{1}{cos(x)}\)
cosecant:
\(csc(x) = \frac{1}{sin(x)}\)
cotangent:
\(cot(x) = \frac{1}{tan(x)}\)

The last ones that I'm going to cover now are the inverse trigonometric functions.  They basically take the ratio of two sides of a triangle and return an angle.  In other words they do the opposite of their inverse,
\(Sin^{-1}(\frac{a}{c}) = Z\)
\(Cos^{-1}(\frac{a}{c}) = X\)
\(Tan^{-1}(\frac{a}{b}) = Z\)
These functions go by many names, but the ones I hear most often sound like "inverse tangent" or "arcsine".  Just stick either an inverse or an "arc" in front of the original function name and you're good to go!  That's all for now on trigonometry.  The potential homework here is enormous (there is a lot of fun you can have playing with trigonometric functions).

Homework:

• Prove that for any \(x\), \(sin^{2}(x) + cos^{2}(x) = 1\).  (Hint: try starting with the pythagorean theorem).
• Convince yourself that for any \(x\), \(\frac{cos(x)}{sin(x)} = tan(x)\).
• Convince yourself that \(sin(\frac{\pi}{2}-x) = cos(x)\)

-Lane