Introduction
Well it is nearly one in the
morning and I have tons of work to do and a fabulous idea pops into my
head: How about writing an introductory tutorial to trigonometry! I am
going to fall so far behind. And once again I did not have the chance to
proof read this or check my work so if you find any mistakes e-mail me.
I'm going to try
my best to write this as if the reader has no previous knowledge of math
(outside of some basic Algebra at least) and I'll do my best to keep it
consistent. There may be flaws or gaps in my logic at which point you can
e-mail me and I will do my best to go back over something more specific.
So let's begin with a standard definition of trigonometry:
trig - o
- nom - e - try n. - a branch of mathematics which deals with relations
between sides and angles of triangles
Basics
Well that
may not sound very interesting at the moment but trigonometry is the most
interesting forms of math I have come across…and just to let you know I do
not have an extensive background in math. Well since trigonometry has a
lot to do with angles and triangles let's familiarize ourselves with some
fundamentals. First a right triangle: 
A right triangle
is a triangle that has one 90-degree angle. The 90-degree angle is denoted
with a little square drawn in the corner. The two sides that are adjacent
to the 90-degree angle, 'a' and 'b', are called the legs. The longer side
opposite of the 90-degree angle, 'c', is called the hypotenuse. The
hypotenuse is always longer than the legs. While we are on the subject
lets brush up on the Pythagorean Theorem. The Pythagorean Theorem states
that the sum of the two legs squared is equal to the hypotenuse squared.
An equation you can use is:
c^2 = a^2 + b^2
So lets say we
knew that 'a' equaled 3 and 'b' equaled 4 how would we find the length of
'c'…assuming this is in fact a right triangle. Plug-in the values that you
know into your formula:
c^2 = 3^2 + 4^2
Three squared plus
four squared is twenty-five so we now have this:
c^2 = 25 - - -
> Take the square root of both sides and you now know that c =
5
So now we are passed some of the relatively boring parts. Let's
talk about certain types of right triangles. There is the 45-45-90
triangle and the 30-60-90 triangle. We might as well learn these because
we'll need them later when we get to the unit circle. Look at this picture
and observe a few of the things going on for a 45-45-90
triangle:
In a 45-45-90 triangle you have a 90-degree angle and
two 45-degree angles (duh) but also the two legs are equal. Also if you
know the value of 'c' then the legs are simply 'c' multiplied by the
square root of two divided by two. I rather not explain that because I
would have to draw more pictures…hopefully you will be able to prove it
through your own understanding. The 30-60-90 triangle is a little but
harder to get but I am not going into to detail with it…here is a
picture:
You now have one 30-degree angle, a 60-degree angle,
and a 90-degree angle. This time the relationship between the sides is a
little different. The shorter side is half of the hypotenuse. The longer
side is the hypotenuse times the square root of 3 all divided by two.
That's all I'm really going to say on this subject but make sure you get
this before you go on because it is crucial in understanding the unit
circle…which in turn is crucial for understanding trigonometry.
Trigonometric Functions
The entire subject of
trigonometry is mostly based on these functions we are about to learn. The
three basic ones are sine, cosine, and tangent. First to clear up any
confusion that some might have: these functions mean nothing with out a
number with them i.e. sin (20) is something…sin is nothing. Make sure you
know that. Now for some quick definitions (these are my own definitions…if
you do not get what I am saying look them up on some other website):
Sine - the ratio of the side opposite of an angle in a right
triangle over the hypotenuse.
Cosine - the ratio of the side
adjacent of an angle in a right triangle over the hypotenuse.
Tangent - the ratio of the side opposite of an angle in a right
triangle over the adjacent side.
Now before I go on I should also
say that those functions only find ratios and nothing more. It may seem
kind of useless now but they are very powerful functions. Also I am only
going to explain the things that I think are useful in Flash…I could go
off on some tangent (no pun intended) on other areas of Trigonometry but
I'll try to keep it just to the useful stuff. OK lets look at a few
pictures:
Angles are usually denoted with capital case letters so
that is what I used. Now lets find all of the trigonometry ratios for
angle A:
sin A = 4/5
cos A = 3/5
tan A = 4/3
Now it
would be hard for me to explain more than what I have done, for this at
least, so you are just going to have to look at the numbers and see where
I got them from. Here are the ratios for angle B:
sin B = 3/5
cos B = 4/5
tan B = 3/4
Once again just look at the
numbers and reread the definitions to see where I came up with that stuff.
But now that I told you a way of thinking of the ratios like opposite over
hypotenuse there is one more way which should be easier and will also be
discussed more later on. Here is a picture…notice how I am only dealing
with one angle:
The little symbol in the corner of the triangle is a
Greek letter called "theta"…its usually used to represent an unknown
angle. Now with that picture we can think of sine, cosine and tangent in a
different way:
sin (theta) = x/r
cos (theta)= y/r
tan
(theta)= y/x -- and x <> 0
We will be using that form most of
the time. Now although I may have skipped some kind of fundamentally
important step (I'm hoping I did not) I can only think of one place to go
from here: the unit circle. Becoming familiar with the unit circle will
probably take the most work but make sure you do because it is very
important. First let me tell you about radians just in case you do not
know. Radians are just another way of measuring angles very similar to
degrees. You know that there are 90 degrees in one-quarter of a circle,
180 degrees in one-half of a circle, and 360 degrees in a whole circle
right? Well if you are dealing with radians there are 2p radians in a
whole circle instead of 360 degrees. The reason that there are 2p radians
in a full circle really is not all that important and would only clutter
this "tutorial" more…just know that it is and it will stay that way. Now
if there are 2p radians in a whole circle there are also p radians in a
half, and p/2 radians in a quarter. Now its time to think about splitting
the circle into more subdivisions than just a half or quarter. Here is a
picture to help you out:
If at all
possible memorize those values. You can always have a picture to look at
like this one but it will do you well when you get into the more advanced
things later on if you have it memorized. However that is not the only
thing you need to memorize. Now you need to know (from memory if you have
the will power) the sine and cosine values for every angle measure on that
chart.
OK I think I cut myself short on explaining what the unit
circle is when I moved on to explaining radians. For now the only thing we
need to know is that it is a circle with a radius of one centered at
(0,0). Now the really cool thing about the unit circle is what we are
about to discuss. I'm going to just pick some random angle up there on the
graph…let's say…45 degrees. Do you see that line going from the center of
the circle (on the chart above) to the edge of the circle? That point at
which the line intersects the edge of the circle is very important. The
"x" coordinate of that point on the edge is the cosine of the angle and
the "y" coordinate is the sine of the angle. Very interesting huh? So lets
find the sine and cosine of 45 degrees ourselves without any calculator or
lookup tables.
Well if you remember anything that I said at the
beginning of this tutorial then you now know why I even mentioned it. In a
right triangle if there is an angle with a measure of 45 degrees the third
angle is also 45 degrees. And not only that but the two legs of the
triangle have the same length. So if we think of that line coming from the
center of the circle at a 45-degree angle as a right triangle we can find
the x- and y-position of where the line intersects…look at this
picture:
If we apply some of the rules we learned about 45-45-90
triangles earlier we can accurately say that:
sqrt (2)
sin 45 = --------
2
sqrt (2)
cos 45 = ----------
2
Another way to think of sine is it's the distance from the x-axis to
the point on the edge of the circle…you can only think of it that way if
you are dealing with a unit circle. You could also think of cosine the
same way except it's the distance from the y-axis to the point on the
border of the circle. If you still do not know where I came up with those
numbers look at the beginning of this tutorial for an explanation of
45-45-90 triangles…and why you are there refresh yourself on 30-60-90
triangles because we need to know those next.
Now lets pick an
angle from the unit circle chart like 30 degrees. I'm not going to draw
another picture but you should know how to form a right triangle with a
line coming from the center of the circle to one of its edges. Now
remember the rules that governed the lengths of the sides of a 30-60-90
triangle…if you do then you can once again accurately say that:
1
sin 30 = ----
2
sqrt (3)
cos 30 = ---------
2
I was just about to type out another explanation of why I did this but
it's basically the same as what I did for sine just above. Also now that I
am rereading this I am seeing some things that may cause confusion so I
thought I would try to clear up a few things. If you look at this picture
(it's the same as the one I used a the beginning of all this) I will
explain with a little bit more detail on how I arrived at those values for
sine and cosine of 45-degrees:
Our definition
of sine states that the sine of an angle would be the opposite side of the
triangle divided by the hypotenuse. Well we know our hypotenuse is one
since this a unit circle so we can substitute a one in for "c" and get
this:
/ 1*sqrt(2) \
| ------------ |
\ 2 /
sin 45 = -------------------
1
Which even the most basic understand of Algebra will tell us that the
above is the same as:
sqrt (2)
sin 45 = --------
2
Now if you do not get that look at it really hard until it comes to
you…I'm sure it will hit you sooner or later. And instead of my wasting
more time making a complete unit circle with everything on it I found this
great link to one: http://www.infomagic.net/~bright/research/untcrcl.gif
. Depending on just how far you want to go into this field of math as well
as others like Calculus you may want to try and memorize that entire
thing. Whatever it takes just try your best. I always hear people talking
about different patterns that they see which helps them to memorize the
unit circle, and that is fine but I think it makes it much easier to
remember if you know how to come up with those numbers…that's what this
whole first part of this tutorial was mostly about.
Also while on
the subject I might as well tell you about the reciprocal trigonometric
functions. They are as follow:
csc (theta) = r/y
sec (theta) =
r/x
cot (theta) = x/y
Those are pronounced secant, cosecant,
and cotangent. Just think of them as the same as their matching
trigonometric functions except flipped…like this:
sin (theta) =
y/r - - - > csc (theta) = r/y
cos (theta) = x/r - - - > sec
(theta) = r/x
tan (theta) = y/x - - - > cot (theta) = x/y
That makes it a little bit easier to understand doesn't it?
Well believe it or not that is it for an introduction to
trigonometry. From here we can start to go into much more complicate
areas. There are many other fundamentals that I would have liked to go
over but this has gotten long and boring enough as it is. I guess I am
hoping that you will explore some of these concepts and ideas on your
own…you will gain much more knowledge that way as opposed to my sloppy
words.
Before I go…
Before I go I want to just give you
a taste of what is to come…this may actually turn out to be just as long
as the above so go ahead and make yourself comfortable. First I want to
introduce to you trigonometric identities, which are trigonometric
equations that are true for all values of the variables for which the
expressions in the equation are defined. Now that's probably a little hard
to understand and monotonous but I'll explain. Here is a list of what are
know as the "fundamental identities":
Reciprocal Identities
1
csc (theta) = ---------- , sin (theta) <> 0
sin (theta)
1
sec (theta) = ---------- , COs (theta) <> 0
cos (theta)
1
cot (theta) = ---------- , tan (theta) <> 0
tan (theta)
Ratio Identities
sin (theta)
tan (theta) = ------------ , cos (theta) <> 0
cos (theta)
cos (theta)
cot (theta) = ------------- , sin (theta) <> 0
sin (theta)
Pythagorean Identities
sin^2(theta) + cos^2(theta) = 1
1 + cot^2(theta) = csc^2(theta)
1 + tan^2(theta) = sec^2(theta)
Odd-even Identities
sin (-theta) = -sin (theta)
cos (-theta) = cos (theta)
tan (-theta) = -tan (theta)
csc (-theta) = csc (theta)
sec (-theta) = sec (theta)
cot (-theta) = -cot (theta)
Now proving them…well that's gonna take a lot of room but here it goes.
I'm only going to prove a few out of each category of identities so maybe
you can figure out the others. Lets start with the reciprocal. Well if the
reciprocal of a number is simply one divided by that number then we can
look at cosecant (which is the reciprocal of sine) as:
1
csc (theta) = ----- ----------------- >>> | If you multiply the numerator and the denominator by "r" you get:
/ y \ |
|---- | < -- I hope you know | csc (theta) = r/y < -- Just like we said before. We just proved
\ r / that is sine (theta) | an identity...I'll let you do the rest of them...
Now the ratio identities. If you think of tangent as y/x , sine as y/r
, and cosine as x/r then check this out:
sin (theta) --- > y/r y
tan (theta) = -------------- --- > ----- --- > Multiply top and bottom by "r" and you're left with --- > ---
cos (theta) --- > x/r x
I'm going to save the proof for the Pythagorean Identities for another
time. These fundamental identities will help us prove much more complex
identities later on. Knowing trigonometric identities will help us
understand some of the more abstract things…at least they are abstract to
me. Once I am finished with this I am going to write another tutorial that
will go into the somewhat more complex areas that I know of and these
fundamental things I have just talked about are required reading.
I
was going to go over some laws that can be very useful but my study plan
tells me that I may not have provided enough information for you to
understand it…therefore that will be something coming in the next thing I
write.
Closing thoughts
Well this concludes all the
things that you will need to know before you start to do more complicated
things. I was a bit brief with some things so if you have any questions or
if you want me to go back and further explain something I implore you to
e-mail me and I will do my best to clear up any confusion. Also I want to
reiterate that this is a very basic introduction to trigonometry. I hope
you were not expecting to read this and learn all there is to know.
Actually I have not really even mentioned Flash or the possibilities
yet…and quite honestly there is not really anything to work with yet.
However once I do start to mention Flash and the math that it will take to
create some of these effects everyone sees it will almost be just like a
review. When you sit down and want to write out a script it will be like
merely translating everything you learned about trigonometry from a piece
of paper into actionscript.
If you want a little synopsis of what
I plan on talking about in the next few things I write here you
go:
- Trigonometry curves
- More advanced look into
trigonometry
- Programmatic movement using trigonometry
-
Orchestrating it all into perfect harmony (pardon the cliché)
Well
that's it for me…until next time.