Intro to the SINe

During my never-ending quest for the truth about Time, which has taken me (meaning Selraybob, since I’m just helping edit) to relativity and quantum mechanics and Heisenberg, I ran across a lot of equations. Some I’d seen before, but a whole bunch, maybe most, were all cryptic and confusing and looked more like ancient Greek than English. Most weren’t even close to what I’d seen in my high school math books, which, I admit, I opened as little as I could. So I had to do some learning. A lot of learning.

[ Next > SIN: From Triangles to Angles ]

Now I’m sharing some of what I learned. These posts aren’t complete math lessons. They’re all focused on the pieces I had to learn so I wasn’t so zipper-down confused every time I opened a book. Mostly, they’re the pieces I needed to learn something about all the misguided talk about Time being some mystical ether. Well, actually, they all lead up to the wave equation and uncertainty–eventually.

So here goes, starting with sine.

Trigonometry

Sine, I found out, is a trigonometric function. A lot of good that did me since I didn’t really know what trigonometry was. I knew the word–it was a course in school–but not much more than that. So I looked it up.

This is what dictionary.com said.

the branch of mathematics that deals with the relations between the sides and angles of plane or spherical triangles, and the calculations based on them.

That was fine. But how triangles and wave equations fit made no sense. One has pointy angles and one is all smooth. But trigonometry’s there, in the middle of it. And intimidating as heck. With a cute little nickname too–trig. As if that makes it less scary. I walked past the trig shelf in the library a good dozen times, avoiding, until finally my buddy Carl dropped a worn old book on my desk and told me to read.

I did. And it was pretty cool once I got into it. Luckily, we don’t need to know all of trig in order to get to the wave equation. We just need the sine and cosine, and they are both based on right triangles.

Right Triangle

A right triangle is a triangle with one right angle (90 degrees), as pictured below.

The letters a, b, and c represent the lengths of the sides. The square indicates a right angle. The capital letters, A and B, identify the two acute (<90 degrees) angles.

The Trig Equations

Each of the trig functions is based on a specific angle, like B or A. So we have to pick one. We’re going to start with angle B.

By definition the sine is the ratio of the length of the segment opposite our chosen angle (b) divided by the length of the segment opposite the right angle, called the hypotenuse (c).

The cosine is similar, but it’s the ratio of the length of the side adjacent to our chosen angle divided by the length of the hypotenuse.

Using angle B, we get:

sine B = b / c
cosine B = a / c

In words, it’s:

Sine = Opposite over Hypotenuse, and
Cosine = Adjacent over Hypotenuse

With numbers, and using the famous 3,4,5 right triangle, we have:

The sine B = 3/5 = 0.6.
The cosine B = 4/5 = 0.8.

For angle A, the sine = 4 / 5 = 0.8, and the cosine = 3 / 5 = 0.6.

It’s also okay to flip the triangle and change the vertical and horizontal sides, and even change which angle is B and which is A.

For the new angle B, the sine = 4 / 5 = 0.8, and the cosine B = 3 / 5 = 0.6.

It’s arbitrary which angle we pick, but once we pick, we’re stuck. The opposite becomes the opposite and the adjacent becomes the adjacent, for as long as we use this triangle.

Proportions

Something I found to be important is that the sine and the cosine don’t depend on the lengths exactly. They only depend on the shape of the triangle.

If all three corners have the same angles, and one of them is right, then the trig ratios will be the same.

I took the famous triangle with the sides of 3, 4 and 5. Then I doubled the straight sides. It’s pretty obvious that if you change every line by the same factor the ratios will be the same.

The sines are 3 / 5 = 0.6 and 6 / 10 = 0.6. They’re the same. The cosines are 4 /5 = .08 and 8 / 10 = 0.8. The same again.

Pythagorean Theorem

The lengths of the sides of right triangles follow a special relationship. The square of the length of the hypotenuse equals the sum of the squares of the two sides. It’s easier with the formula.

c² = a² + b²

Or:

c = sqrt (a² + b²)

If the structure of the triangle is the same, and all sides are in the same proportions, then the angles will all be the same. In fact, all we need to define a triangle is one line segment, a right angle, and one other angle. And because for our trig functions, triangles only have to be proportional, all we really need is an angle. From that we can get the rest.

Proportional Revisited

Since I knew the 3,4,5 triangle was special, I decided to check the proportional thing for other triangles. So, using the Pythagorean Theorem, I created a little spreadsheet. I used 1.2 as the factor, and three and six and 2.25. Multiplying each side by the same factor, then using the Pythagorean Theorem, I found that the sine and cosine remained the same. This matters.

The Quadrants

If you’ve seen the coordinate axes, you’ve probably heard about the quadrants and know what the origin is. They’re basic, but I numbered them just as a reminder.

And, according to the conventions of math, the right side of the y axis is positive for the x value. To the left is negative.

And above the x axis is positive for the y value. Below the axis is negative for the y value.

This is basic, I know. But it matters for the sine and cosine, because as we start merging the ideas of triangles with the conventions of the coordinate axes, the quadrant can determine whether the sine is positive or negative.

Trig and the Quadrants

When mathematicians draw pictures of the angles on a set of axes, they connect what we’re going to call the angle-line–the green ones below–to the origin. This makes the math easy. It wouldn’t change anything real if the angle-line crossed somewhere away from the origin, but the numbers would be harder to work with. So they put the start of the angle-line at (0,0)

To make things standard, and to have some meaning, all the angles we’re going to worry about are the ones made with the x axis. In order to get our sine and cosine values from the angle-lines, we have to create triangles. All the triangles are going to be constructed by connecting the angle-line (anywhere on the line, since the triangles are proportional) to the x axis with a vertical line. That will create a 90 degree angle. This will create the four right triangles as pictured below. The green line, which we’ve call the angle-line, is the hypotenuse of each triangle.

The angles made with the x axis are the B’s. This means that the opposite sides, b, will be the length of the vertical side, as shown. And the adjacent sides, a, will be along the x axis. However, the values of those sides will follow the conventions related to the axes. So the adjacent side, a, will be positive to the right of y and negative to the left. The side b will be positive above the x axis and negative below.

However, based on the Pythagorean Theorem, and standards, he hypotenuse, c, will always have a positive value.

Now for a couple examples with numbers.

In this case, with the triangle in quadrant 2, the sine is 12/13 (a positive number) and the cosine is -5/13 (a negative number.)

In the 3rd quadrant case, the sine is -3/5 (negative) and the cosine is -4/5 (also negative).

Going back to our red and blue arrows.

Quadrant 1 (0^o – 90^o) — Both positive
Quadrant 2 (90^o – 180^o) — Sine is positive, cosine is negative
Quadrant 3 (180^o – 270^o) — Both negative
Quadrant 4 (270^o – 360^o) — Sine is negative, cosine is positive

And when the angles exactly line up with 0, 90, 180 or 270, the values are either 1 or zero, depending on which you’re calculating.

Summary

Here’s what we have so far.

1. The equations:
   1. Sine = opposite over hypotenuse
   2. Cosine = adjacent over hypotenuse
2. As long as the angles are the same, the triangles are proportional. The resulting sines and cosines will be the same regardless of the size of the triangle.
3. When working with the coordinate axes and an angle-line, the triangle used to determine the trig functions is made by connecting the angle-line, which becomes the hypotenuse, with the x axis using a perpendicular line.
4. The hypotenuse is always positive, and follows the Pythagorean Theorem. The other sides can be positive or negative, depending on where the triangle is located relative to the coordinate axes we’re using.

This is only triangles–not anywhere close to waves. So the next post will explore how these triangle measurements relate to angles, and from there we’ll get into waves.