I'm taking Trig next semester and I was wondering if there were any free online courses that someone could recommend to me. I want to learn it so I don't really have to do shit while I'm actually in the class. Also, this is unrelated to this section, but if you could also recommend a similar program for chemistry that would be much appreciated. Thanks, Nin'
Actually wikipedia has good links, Have you tried typing "online trigonometry courses" into google? The other option is getting a good textbook. Trig is usually introduced at high school level. Circular functions are important, for some reason or other (you may actually find out why, but they don't tell you much for quite a while).
all you need to know is basically that \(\sin ^2(\theta) + \cos^2(\theta) = 1\). The rest is pretty much derived from that equation. Unless we start talking Lobachevskyian geometry. geometry is super easy, dont sweat it. If you need help on a problem just post it here and I'll hep.
Keys to understandingTrig I teach HS Math. Trigonometry at first looks like the study of "metering/measuring triangles" It is. But it has profound and deep significance in Mathematical descriptions of the natural world and Physics because the Trig functions of sine and cosine describe ALL periodic phenomena. The beating of your heart, your breathing, blinking of your eyes, the motion of earth around the sun, a bouncing ball, etc. All physical system responses to stimulii, both transient and steady state are described by trig functions. They key to seeing this transition from triangles to oscillitory/harmonic/periodic motion is with the Unit Circle. Study that and understand that the coordinates of all points on the circle are ( cosine x , sine x ) where x is the angle between the horizontal line [ x axis ] and a line thru the point given point on the circle from the center. The radius of the Unit Circle is unity or 1. Then you will see that the circular motion of of the point around the circle is described by sinusoids. Cheers
For example, I thought all of trigonometry was: \(e^{i x} = \cos x + i \sin x\) From this you get the Taylor series for sin and cos. And from that you get: \(\begin{eqnarray} \cos x & = & \cos (-x) \\ \sin x & = & - \sin (-x) \\ \cos x & = & \frac{e^{i x} + e^{- i x}}{2} \\ \sin x & = & \frac{e^{i x} - e^{-i x}}{2 i} \\ \cos^2 x & = & \frac{e^{2 i x} + 2 + e^{- 2 i x}}{4} = \frac{1 + \cos 2x}{2} \\ \sin^2 x & = & \frac{e^{2 i x} - 2 + e^{- 2 i x}}{-4} = \frac{1 - \cos 2x}{2} \\ \sin^2 x + \cos^2 x & = & 1 \\ \cos 2x & = & 2 \cos^2 x - 1 = \cos^2 x - \sin^2 x \\ \dots & & \dots\end{eqnarray} \) But there are some fundamental things to learn about the triangles which are also fundamental and useful if you wish to know how far away the stars are. YouTube: History of the Universe Made Easy http://www.astro.ucla.edu/~wright/distance.htm
Don't forget to memorize the "unit circle." I thought \(e^{i x} = blah\) was derived from the Taylor series for cos and sin and then we simply set \(e^{i x}\) to be defined that way...? Well, it doesn't really matter which way you do it... in either direction one goes, we get the same thing.
I think \(e^{ix}\) is not directly defined by the Taylor series of cos and sin, but when you analytically continue the real exponential function to the complex plane you happen to get this formula with cos and sin. There must be a deeper explanation. Anyone?
There's this law of equal areas connected to planetary orbits that's a kind of link to calculus, Newton and all that, how he found you could fit a triangle in there, the parallax of nearby stars that observations confirm means we're in relative motion with too.
The way I learned it is that they looked at the Taylor series for e^x where x is a real number, replaced x with i*x, commuted some terms, factored out i, and saw plain-as-day the taylor series for cos and taylor series for sine... then we simply defined e^{x i} to be what we have now.
That's exactly what Temur said - you aren't defining \(e^{i x} = \cos x + i \sin x\). It drops out when you analytically continue the exponential function to complex arguments. Maybe a semantic point but it's important nonetheless.
