Round and Round – trig functions

Last week we talked about some basic functions that you will deal with in algebra, calculus and beyond. Today I am going to introduce you to some basic trig functions.

One of the most fascinating applications of trig functions is that of daylight hours. Do you notice how the days get longer in the summer, then shorter in the winter? And then they get longer and shorter and longer and shorter, and the cycle just keeps going on.

Trig functions are just that – cycles.

This is a graph of the function f(x) = sin(x) (the one on the left is one period of the sine function, and the one on the right shows more what the graph does – it just keeps going over and over again – in a cycle.)

 

This is a graph of the function f(x) = cos(x). Cosine is a lot like sine – it just starts in a different place. Where sine starts at zero when x = 0, cosine starts at 1 when x = 0.

This is a graph of the function f(x) = tan(x). Tangent is a ratio of sine and cosine. The reason it is undefined at some places (see how the graph goes up and doesn’t come back down, and then it stars from far below?) is because sometimes cosine is zero, and you can’t divide by zero.

This is just a little introduction to show you what trig functions look like. There are also inverse trig functions, which we’ll talk about later. We’ll also talk about some really interesting applications of trig functions.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.

Sequences Part I - One After the Other

 

Sequence.

Say it with me: seee-kwence.

Sequences can look scary, and can make absolutely no sense. What are these lists of numbers that march on toward infinity? Let's break it down a little.

For those who are mathematically challenged, defining math terms in terms of math (especially to someone who doesn't "speak" math fluently) is a lot like using a "circular definition" for example, this one:

Hill - "1: a usually rounded natural elevation of land lower than a mountain" [1]

Mountain - "1a: a landmass that projects conspicuously above its surroundings and is higher than a hill"

So, in an effort not to be circular, I will define the word sequence as you would in your English class before I jump into the math part.

Here are a few definitions from Merriam-Webster to get you "in the mood":

2 : a continuous or connected series: as

b : three or more playing cards usually of the same suit in consecutive order of rank

f (1) : a succession of related shots or scenes developing a single subject or phase of a film story (2) : episode

3 a : order of succession

And here are some examples of some (non-math-related) sequences:

1. He listened to the telephone messages in sequence.
2. a chase sequence in a spy movie
3. I enjoyed the movie's opening sequence.

I hope this gives you a better feeling for the actual word sequence, which I think will help you understand the math variety, too. So, a sequence is just putting things in order. I'm sure you can think of a lot of things that we put "in order" - underwear, then pants; socks, then shoes; turn on the car, then put it in drive. You can probably think of some longer ones, too - like, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. Those are all sequences. Groups of things that go "one after the other" in a certain order.

The place where math sequences get scary is because they are infinite. That means they never end. That's like if you had to start walking and never ever ever stop. You would just keep putting one foot in front of the other, and walk forever.

Well, fortunately, we have a nice sequence we all know (some of us know more of it than others) and we can use it to talk about all the other sequences. This "nice sequence" is called the integers.

We have all been learning about integers since we were little kids. "Jump on three, okay? One, two, three!" Those are positive integers (the integers actually go all the way backward in the negative "direction", too).

You might remember that there is another set called the natural numbers - we also call those the "counting numbers." The only reason we are using integers instead of natural numbers is because the integers include the amazing 0, and we really need that one.

So, let's look at our “nice sequence” – the positive integers. We will use this set of numbers to talk about all the other sequences.

0,1,2,3,4,5,6,7,.... you get the picture, right?

Well, now we just use a letter (*gasp!* I know... letters make everything confusing). But before you start hyperventilating, think of our letters as little cups.

We will represent our sequence with , which is just like the cup on the left. We have “n” cups, which just means for every number (“n”) in the positive integers (0,1,2,3,4,5,6,7,…..), we will have something in our cup. The way we figure out what number goes in our cup is by a function. There are basically two types of functions for sequences. Just your basic function, like , or a recursive function (which just means each number in the sequence depends on the numbers that came before it), like , which happens to be the Fibonacci sequence.

Now, to find out what we put in each up, we take the number that is on the cup and plug it into our “recipe” (the function for the sequence).

Let’s do an example. Say we have the sequence . Let’s fill up the first five cups. For the first cup, 0, we plug zero into the function (2n) and we get 0, so the first cup is 0 (in other words, = 0). For the second cup, we plug 1 into the function and get 2, so we put 2 inside the second cup (in other words, = 2). If you keep doing this for all the cups, you’ll find out that cup #2 = 4, cup #3 = 6, cup #4 = 8 and cup #5 = 10 (in other words, = 4, = 6, = 8, and= 10). Now, we can line up the sequence “one after the other” and see 0 , 2 , 4 , 6 , 8 ,10 ,… which I hope you recognize as the even numbers. To write this sequence down in “math” it would look like this:

That is how you would write the sequence in “math.”

Please leave comments on how well you understood this concept.

Tomorrow… Sequences Part II – The Future is in the Past

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.