Trigonometry:
Triangles,
Circles, and Waves
An
Overview of Trigonometry
Copyright
(c) 1996 by Kenny Felder
This sheet
is meant as a guide to help you review trigonometry. I assume
you're already familiar with the basic concepts, and just need
to have things cleared up or refreshed; if you've never seen
trig before, this sheet will probably only be helpful in
conjunction with a class.
The
Basic Idea
Start out
by considering a right triangle: that is, a triangle with one
90o angle. The angle that we care about is one of
the other ones, which we'll label with the Greek letter q. (Everyone
always labels angles with Greek letters. Don't ask me why.)
We're going to look at the sides: the big one, opposite the
right angle, we call the hypotenuse. The other two don't
have special names, but one is adjacent to q and one
is opposite q, so those will do. In the triangle below,
for instance, the side opposite q has length 2 (inches, or meters,
or whatever). The side adjacent to q has length .
The hypotenuse, therefore, must have length 4 by the Pythagorean
theorem.
Now, we're
going to start by defining the sine of q (written "sin q")
as the length of the opposite side, divided by the length of
the hypotenuse; so we'll say that sin 30o = (2
divided by 4) = 1/2. You might well ask why we would
give a special name to such a thing. I'll be talking soon about
how it can be practical and useful, but the key point to understand
is that the sine of q depends only on q; it does not depend
on the specific triangle. If you change the size of the
triangle I drew up there, but leave the angles the same, the
lengths of the sides will change, but the ratio
sin q = opp/hyp will remain the same. So sin q
gives us a way of representing the fact that all triangles that
have the same angles, will have the same side ratios.
Look at
the drawings below and see if you can convince yourself that
sin q is the same for different triangles with the same
q.
Before I
work an example in which the sine proves useful, let me mention
the other two important trig functions. The cosine of q (write
"cos q") is the adjacent side divided by the hypotenuse,
and the tangent ("tan q") is the opposite side over the
adjacent. You can memorize these by remembering the indian-sounding
word SOHCAHTOA: it doesn't actually mean anything, except
"Sine = Opp/Hyp, Cosine = Adj/Hyp,
Tangent = Opp/Adj," but it is
a hard word to forget if you say it out loud a few times. (Note:
A couple of readers have pointed out alternative mnemonics:
"Oh Heck, Another Hour Of Algebra" or "Oscar Had A Heap Of Apples."
These are different tricks for memorizing the same thing: Opposite
over Hypotenuse, Adjacent over Hypotenuse, and Opposite over
Adjacent, for sin, cos, and tan, respectively).
Test yourself:
looking at the triangles drawn above, what are the cosine and
tangent of 30o? Check your answers on a calculator:
punch in 30 and then hit the cos button to get
cos 30, and so on.
Knowing
the meanings of the sine, cosine, and tangent, you can work
most elementary trig problems if you have a calculator or trig
tables. For instance, a typical problem might be: you want to
measure the height of a building without climbing it. So, you
stand thirty feet away from the building, and you measure an
angle of elevation of 60o to the building top. How
tall is the building?
To start
any such problem, draw a picture. Make sure you label all the
relevant angles and sides.
Now, look
at which sides are important. In this case, the side you know
is adjacent to the known 60o angle; and the
side you want to know is opposite the angle. This tells
you which trig function you want: the tangent, since it relates
the opposite and adjacent sides and ignores the hypotenuse.
From here on, it's easy. Use a calculator or trig table to tell
you that tan 60o = 1.73. Then write tan q
= opp/adj and you can write the equation 1.73 = ht/30'. Solving,
ht = 51.9 ft. Without leaving the ground, you have measured
the height of a building very exactly: simply by knowing that
the ratio tan q = opp/adj is the same for a gigantic
building-person-cement triangle as for the little computer-drawn
triangle on the page here! If you understand what we've done
so far, you've got the basics; from here on in, it's just refinements.

Trig
Identities
Something
most courses spend a lot of time on are the so-called "trig
identities": that is, formulas that relate the various trig
functions to each other. For instance, looking at the definitions
sin q=opp/hyp, and cos q=adj/hyp, what is the ratio
sin q/cos q? A little algebra will show you that this
ratio equals opp/adj, which is just we've already defined as
tan q! (Make sure you check the algebra here yourself;
don't take my word for it.) So we have a trig identity: tan q
= sin q/cos q. What makes this a trig identity
is that it holds for any angle q; you might want to check this
for yourself on a calculator for a few different angles.
The other
always-useful trig identity is that if you take sin q and square
it; and then you take cos q and square it; and you add these
two things; you always get one. You might think this would be
written like this:
(sin q)2+(cos q)2=1
But in fact,
it is generally written like this instead:
sin2q+cos2q=1
So sin2q
is just the way that people tend to write "sin q, squared,"
for some reason.
Anyway,
I would encourage you to check this one out on a calculator
too. Pick a number like 37, and type the following into your
calculator.
37
sin x2 + 37 cos x2 =
and see
if you always get 1. Then, for real fun, figure out why
this particular relationship is true for any angle; this is
pretty easy, using the definition of the sin and cos, and the
pythagorean theorem. (I know, I know, you don't like doing proofs;
but try this one anyway, honest.)
There are
a million other trig identities, which enable you to rewrite
sin(2q) or whatever; on the whole, you memorize these for the
test and then forget all about them. Lord knows, that's what
I do.

The
Unit Circle
The triangle
picture of trig is very useful, but once we have it, we can
discuss the trigonometric functions in a number of other useful
ways. The two most popular are the unit circle, and the sine
and cosine graphs. Each picture can be used to tell you different
things about the trigonometric functions, but it's important
to remember that all three pictures are describing the same
sine, cosine, and tangent. So to start with, let's look at the
unit circle.
This drawing
shows the unit circle; that is, a circle around the origin,
with radius one. It also shows a line going from the origin,
to some point on the circle. The angle from this line, to the
x-axis, is labeled q. And there is a point, where the line
crosses the circle, which I have labeled (x,y).
I've also
drawn one other line, which goes from (x,y) straight down to
the x-axis. I drew this vertical line because it gave me a little
triangle. Now, based on our earlier triangle definitions, what
is cos q? Well, cos q is the adjacent side over the
hypotenuse, so look at the triangle in the drawing. The adjacent
side is the x-coordinate of our point (x,y), which is just x.
The hypotenuse is the radius of the circle, which is one. So
the cosine is x/1, or x. You can just as easily prove to yourself
that sin q is y. So, using the triangle definitions we
discussed earlier, we have come to the circle definition of
the trig functions:
Draw a line
from the origin, at an angle q, and look at the point where
that line intersects the unit circle. cos q is the x-coordinate
of that point, and sin q is the y-coordinate.
This is
a new definition of our trig functions; you can do everything
based on this circle-definition instead of the triangle-definition,
if you want to. But it should be clear from the above discussion
that they are really the same thing.
What may
not be clear is why you would want to have two different ways
of defining the same thing, even if they do wind up being the
same! The triangles were fine, why do we need circles? I think
the biggest reason is that in our earlier definition, q was
one angle of a right triangle, and such angles are always
bigger than 0, and smaller than 90o. But on the
circle, q can be anything: it can be much bigger than 90o,
or it can even be negative!
For instance,
what's sin 180o? Well, if you look at the 180o
angle on the circle, it intersects at the point (-1,0). So cos 180o
= -1 and sin 180o = 0.
When you
get used to this, it can be a great time saver and boost your
intuitive feel for the trig functions enormously. As a small
test for yourself, what are the sine and cosine of 90o?
(Check your answers on a calculator!) With a little practice
you can do this one and a few other easy angles in your head,
just by picturing the unit circle.
As another
example of the usefulness of the unit circle, consider the sine
and cosine of (-q). Well, on the unit circle below I've drawn
in an angle q and its opposite (-q). You can see that the x-coordinates
(the cosines) are the same and the y-coordinates (the sines)
are opposite, so we get the trig identities:
cos
(-q) = cos q and sin (-q) = -sin q.
Again, you
might want to check these identities yourself using a calculator
and a few simple angles. For instance, find the sine and cosine
of 10o, then of -10o. Are the two cosines
the same? Are the two sines opposite?
One more
example of the usefulness of the unit circle: textbooks often
give you an angle, and ask you if the sign and cosine are negative
or positive. They encourage you to memorize rules about what
is negative and positive in different quadrants, but there is
really no need to memorize anything: you just need to picture
the unit circle. For instance, suppose the angle is 110o.
Well, you can see on the unit circle that this puts us in the
upper-left-hand quadrant; so x is negative and y is positive;
so cos 110o is negative, and sin 110o
is positive. You can't figure out the exact values without checking
your calculator; but if your calculator gives you a negative
sin, you can bet you punched something in wrong!
Also, remember
that tan q = sin q/cosq. So tan 110o will also be
negative, since a positive number divided by a negative number
will always give you a negative.
As with
most things on the unit circle, this whole train of thought
takes more time to write out than it takes to actually do it.
Take a little time to work with the unit circle, until you can
picture it in your head, and it will be a very great help to
your understanding of the trig functions.

Graphing
the Sine and Cosine
Before you
can fully understand the behavior of any sort of function, you
have to graph it. In the case of the trig functions, the graph
is pretty easy to draw if you look at the unit circle for a
while. The strategy is simple: start out at q=0o,
and follow the unit circle around, watching what the y-coordinate
does (for the sine, that is; watch the x-coordinate for the
cosine). I'm about to do the sine: follow me, around
the unit circle, then try to do the cosine for yourself.
Sin 0o
is zero, as you hopefully figured out for yourself by looking
at the unit circle last page. Sin 90o is 1.
In between, as q moves from zero to ninety, the y-coordinate
on the unit circle is constantly increasing. So the graph starts
out like:
Continuing
to follow the unit circle around, as q increases to 180o
the y-coordinate drops to zero: we're now at the point (-1,0)
on the unit circle. As q keeps going from there, the y-coordinate
doesn't turn around and go back up to one; it keeps going down,
reaching -1 when q gets to 270o. Then the y-coordinate
turns around, and reaches zero again when q is 360o;
we are now, of course, at exactly the same place on the unit
circle that we were at when q was 0o. So the full
sine graph looks like:
You can
keep on going from there around the unit circle, retracing the
path around and around; so the graph repeats itself indefinitely.
If you followed
me carefully around the unit circle, all of that should have
made sense. If you're like most people, you kind of fuzzed out
half way; so do yourself a favor, and do the cosine graph for
yourself. Start at 0o, where the x-coordinate is
1, and walk yourself all the way around the unit circle, following
what the x-coordinate does from 0 to 90o, then to
180o, then to 270o, and finally back around
to 360o, which is the same place on the circle as
zero. See what the graph looks like for one full cycle, and
convince yourself that it will now repeat itself indefinitely.
When we
went from the triangle to the circle, I said that there were
certain things that would be easier to understand with this
new representation. Similarly, the graph gives us ways of understanding
things are harder with both the triangle and the circle. For
instance, look at your sin and cos graphs side by side. Note
that they look exactly the same, except that they are offset
by 90o. This gives us yet another trig identity:
sin q=cos(q-90o). Once again, you don't have
to memorize this identity: you can easily remember it, just
by knowing what the graph looks like.
We're Done!

Summary
What we've
just covered is the three basic ways of looking at trigonometry:
with triangles, circles, and the graph. If I've been successful,
you now have a feeling that...
-
The
triangle representation of trig makes sense. That is, it
makes sense to define these things called sine, cosine,
and tangent, because no matter what triangle you draw with
a 37o angle, the sine of 37 will always be the
same.
-
The
circle representation of trig makes sense. That is, you
can see that it really is another way of representing the
exact same functions that we discussed on the triangle;
and, you can see that there are certain kinds of things
that are easier to understand on the circle than on the
triangle.
-
The
sine and cosine graphs make sense. That is, you can see
that these are yet another way of representing the exact
same functions that we discussed on the triangle and the
circle; and once again, you can see that there are certain
kinds of things that are easiest to understand on the graph.
The next
step is to start working problems. Some of them will require
using trig identities that you just have to look up, and that's
fine. But the basic ideas should all make sense.

A
Bit More Background Added By Suggestion
Mike Housky was kind enough to write to
me and suggest the addition of some background material. The
first paragraph I quote verbatim from him, since it is a very
cool bit of trivia that I did not know, and I don't have anything
useful to add to it.
"One
thing that might help the opening is a discussion of the word
'trigonometry'. 'Trigon' comes from Greek roots meaning 'three
sides' and is another name for a triangle. '-Metry' is a suffix
meaning 'measurement' and the full term roughly translates
to 'triangle measurement'. This could add some motivation—or
perhaps just some added color—to the discussion of the triangle
approach."
In addition,
Mike goes on to suggest that "In the discussion of the unit
circle approach (my favorite, by the way), some mention of radians
seems appropriate." I think this is a great idea, so here I
go!
As you presumably
know, angles are usually measured by a system where a right
angle is 90°; so a straight line is 180°
and the full circle is 360°. As far as I know,
this is completely arbitrary; there is no good reason to call
a right angle "90" of anything.
Radians
are a measure that are not completely arbitrary. When you measure
in radians, a full circle is 2p, so a right angle is p/2. Now,
this may seem like a big step in the wrong direction, but consider
this: the circumfrence of a circle is 2p times the radius. A-ha!
(Yeah, what's
your point?)
The point
is, on our unit circle, consider some angle q.
In the drawing,
I have labeled an angle q, and also the arc l along the
circle from 0 to q. I think you'll agree that if q gets bigger,
l also gets bigger. If q is 0, l is 0. What may
not be quite so obvious, but I hope it makes sense, is that
they are perfectly proportional: if q doubles, l doubles.
If we measure q in radians, then the constant of proportionality
is 1: in other words, the angle is the arc length. Because
if we go around the whole circle, they both go to 2p.
So radians
are just another way of measuring angles, just like degrees
are. But with radians, you get the added bonus that the angle
is the arc length—or, when it isn't a unit circle, the arc length
divided by the radius of the circle. Pretty cool, huh?

Primer:
Read
An in-depth Review of Basic Trigonometry with Kenneth Boyd's
Trigonometry Crash Review
Read what Excel can do for you with a copy of the Cataclysm
SRD-XL Utility help file.
Online:
Kenny
Felder's The
Math and Physics Help Home Page
Kenneth Boyd's Math
page, or the Zaimoni.com
Home page.
Email:
Send comments or questions to the author of this page, Kenny
Felder
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