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### Trigonometry in a Nutshell

```
Date: 04/11/2001 at 02:09:21
From: Jimmy Lee
Subject: Trigonometry

I'm in 8th grade in my school and in my math class we're doing stuff
about trigonometry (sine cosine hypotenuse). But what is trigonometry?
Can you give me some good easy questions in trigonometry so I can try
using it? Thanks.
```

```
Date: 04/11/2001 at 16:36:23
From: Doctor Ian
Subject: Re: Trigonometry

Hi Jimmy,

Believe it or not, I just spent part of last weekend explaining
trigonometry to my mother, who was upset because about 50 years ago
she wanted to become a radio technician in the navy, but ended up as a
pharmacist's assistant because she couldn't get the hang of
trigonometry.  So I'm going to tell you what I told her.

When you have a right triangle,

/|
/b|
C  /  |
/   |  A
/    |
/_a___|

B

there are basically five things that you can know about it: the
lengths of the sides (A, B, and C), and the measures of the acute
angles (a and b). The third angle is always 90 degrees because it's a
right triangle.

If you know two of the sides, you can use the Pythagorean theorem to
find the other side:

A = sqrt(C^2 - B^2)

B = sqrt(C^2 - A^2)

C = sqrt(A^2 + B^2)

And if you know either angle, a or b, you can subtract it from 90 to
get the other one:

a + b = 90

But what if you know the sides and you want to know the angles?  Or
you know an angle and a side and you want to know the other sides?

Well, here is the central insight of trigonometry: If you multiply all
the sides of a right triangle by the same number (k), you get a
triangle that is a different size, but which has the same angles:

/|
/b|
/  |
/|          k*C  /   | k*A
/b|              /    |
C  /  |             /     |              k*A   A
/   |  A         /      |              --- = -, etc.
/    |           /       |              k*B   B
/_a___|          /__a_____|

B                k*B

Why is that interesting?  Well, it means that if you know the _ratio_
of any two sides, it tells you what the angles are.

For example, let's look at a right triangle in which the acute angles
are 30 and 60 degrees:

/|
/ |
/30|
C  /   |
/    |  A
/     |
/_60___|

B

If side B is 1 unit long, then side C is 2 units long, and side A is

A = sqrt(2^2 - 1^2)

= sqrt(4 - 1)

= sqrt(3)

units long. Now, notice that I haven't said what a 'unit' is.  It
could be a mile, or an inch, or 13.5 cm, or the distance from my big
toe to the base of a bookcase on the other side of the room. If I know
that angle a is 60 degrees, then I know that the following must be
true:

the ratio of A to C is sqrt(3)/2

the ratio of B to C is 1/2

the ratio of A to B is sqrt(3)/1

And, just as importantly, if I know that for a particular triangle,
the ratio of A to B is sqrt(3)/1, then angle a _must_ be 60 degrees.
It can't be any other angle.

So, what we can do is write down a table, in which we list all the
ratios for all the angles of interest:

angle a   ratio A/B
-------   ---------
0         0
.         .             <= values for other angles filled in
.         .
30        sqrt(3)/3
.         .
.         .
45        1/1
.         .
.         .
60        sqrt(3)/1
.         .
.         .

Now we can use this table to find the ratio A/B if we know angle a -
just find the angle in the column on the left, and read the ratio from
the column on the right. Or we can use the same table to find angle a
if we know the ratio A/B - just find the ratio in the column on the
right, and read the angle from the column on the left.

But instead of 'ratio A/B for angle (a)', we use the shorter name
'tangent of angle a', or just 'tan(a)'.

Since there are three possibilities for which pair of sides you might
know - A and C, B and C, A and B - we have three different functions:

angle a  =>  sin(a) = A/C

cos(a) = B/C

tan(a) = A/B

So we can make up a table that looks like this:

Machinery Handbook Math Pages: Trig Tables
http://www.industrialpress.com/Trig.htm

Note that we don't really need include the tan, sec, and csc
functions, since we can just compute them from the sin and cos
functions:

tan(a) = sin(a) / cos(a)

sec(a) = 1 / cos(a)

csc(a) = 1 / sin(a)

So, having said all that, the whole _point_ of trigonometry is this:
If you have a right triangle in which you know one side and any other
side or angle, you can figure out the remaining sides and angles
without having to measure them. In fact, here are all the
possibilities:

You know                How you find the others
------------------      -----------------------------------
two sides               Use the Pythagorean theorem to find the
remaining side.
Use the ratios of the sides to find the
angles.

one side and           Use the sin, cos, and tan functions
one angle              to find the remaining sides.
The other angle is just 90 minus
the one you know.

That's it. The rest of it is shortcuts and tricks. For example, if you
have a triangle like

/|
/ |
C   /  |
/   |
/    |  A
/     |
/_17___|

25

it must be true that

A / 25 = tan(17)                      25 / C = cos(17)

A = 25 tan(17)             25 / cos(17) = C

25 * (1 / cos(17)) = C

25 sec(17) = C

where 'sec' is short for 'secant', and sec(x) is the just another way
to write 1/cos(x). Anyway, when you've put in about a thousand hours
of practice, you will be able to just label the triangle immediately:

/|
/ |
25 sec(17)  /  |
/   |
/    |  25 tan(17)
/     |
/_17___|

25

Also, you will be able to use formulas (called 'identities') like

sin(x-y) = sin(x)cos(y) - cos(x)sin(y)

to do tricks like this:

sin(15) = sin(45 - 30) = sin(45)cos(30) - cos(45)sin(30)

= (1/sqrt(2))(sqrt(3)/2) - (1/sqrt(2))(1/2)

= sqrt(3)/[2 sqrt(2)] - 1/[2 sqrt(2)]

= [sqrt(3) - 1] / sqrt(2)

Whether being able to do these things from memory instead of from a
book is _worth_ a thousand hours of your life isn't clear. You'll have
to take that up with your state legislature or the Department of
Education.

Anyway, that's trigonometry in a nutshell.  All you have to do to make
up problems is draw a right triangle:

/|
/b|
/  |
C  /   |  A
/    |
/     |
/_a____|

B

and then make up a story to go with it, in which you know two sides,
or a side and an angle. For example, a ladder is leaning against a
wall at an angle of 70 degrees with the ground. It just touches the
bottom of a window 10 feet off the ground. How long is the ladder?
How far from the wall is the bottom of the ladder?

window
/|
/ |
/  |
/    |
/     |
/_70___|

?

To turn it into a different problem, just change what you know:

window
/|
/ |
/  |
20 ft /   |  ?
/    |
/     |
/_?____|

8 ft

Or, if you get bored with ladders and windows, make up a different
story, e.g., a bird is sitting on a flagpole, and the sun is at an
elevation of 57 degrees. The shadow of the bird is 11 feet from the
base of the flagpole. How tall is the flagpole?

bird
/|
/ |
/  |
/   |  ?
/    |
/     |
/_57___|

11 ft

Or one person is at the top of the Grand Canyon, and another is at the
bottom.  They use a laser range finder to determine that they are 1200
meters apart, and the one below has to look up at an angle of 78
degrees to see the one above. How deep is the canyon?

Making up your own problems will give you a much better feel for why
this is so useful, and with any luck, you'll start to see why some
people even think it's fun.

So that's how you make up problems. You solve problems made up by
other people by reversing the steps. That is, you read the problem and
try to figure out what is at the vertex of each triangle. Then you try
to figure out which sides and/or angles you've been given, and which
one you're supposed to find. Then you use the Pythagorean theorem and
the sin, cos, and tan functions to find it.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/12/2001 at 11:55:41
From: Jimmy Lee
Subject: Re: Trigonometry

Thanks! You really helped me and saved me some money from buying books
about trigonometry. This always was a toughy for me on tests, but I
think I can work it out next time. Thanks again!
```

```
Date: 04/11/2001 at 05:18:58
From: Michael
Subject: Geometry

I am totally confused about sine, cosine, and tangent. What are they
and where do they come from? All I have been taught is when to use
them and where the button is on the calculator. I have tried finding a
beginners' guide or tutorial on the net but everything is too
advanced. They even use sine etc. with circles and waves, where I
thought it was just to do with triangles. Please could you explain
them? I think it would be very useful to have explanations on your
site, as everybody in my class can use them but nobody knows what they
really are.
```

```
Date: 04/11/2001 at 16:41:34
From: Doctor Dyno
Subject: Re: Geometry

Hi Michael,

Sine, cosine, etc. do have to do with triangles, but you can then
extend the concept to circles and waves.  I think the best way to
"see" this is by going to this site:

http://catcode.com/trig/

Actually, if you just find a quiet place and sit down and open
yourself up to these ideas and sloooow down, it should make sense.
My students are in the habit of going through the material very fast,
just to get done.  But, "getting it" and "getting done" are two
different things.

Cheers,

- Doctor Dyno, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 04/11/2001 at 17:19:16
From: Doctor Ian
Subject: Re: Geometry

Hi Michael,

What does trigonometry have to do with circles?  Well, draw a circle
whose center is at the origin of the x-y plane, and whose radius is 1.
Now choose any point on the circle, draw a line segment from the
center, and mark the angle a between the positive x-axis and the line
segment. You should have something that looks like this:

1|.
|      .(x,y)
|      /   .
|     /
|    /        .
|   /
|  /           .
| / a
---------------
1

Guess what?  The coordinates of the point you chose are

(x,y) = (cos(a), sin(a))

Furthermore, since the radius of the circle is 1, the Pythagorean
theorem tells us that

(sin(x))^2 + (cos(x))^2 = 1

Actually, a lot of formulas, like the ones you can find in the Ask Dr.
Math FAQ: Trigonometry Formulas,

http://mathforum.org/dr.math/faq/formulas/faq.trig.html

are easiest to understand if you think about the unit circle rather
than triangles of arbitrary size.

And what does this have to do with waves?  If you start taking
different values of a, and plot the corresponding values of x and y
(that is, if you plot x and y as functions of a, the way you would
normally plot y as a function of x), you get something like this:

x                        x
|   x                 x            x = cos(a)
|     x             x
+------x-----------x------------
|       x         x
|         x     x
x

a=0    90    180  270    360

|     y
|  y     y                         y = sin(a)
|y         y
y-----------y-----------y------
|            y         y
|              y     y
|                 y

a=0     90   180   270   360

Each plot keeps wiggling back and forth between 1 and -1, repeating
itself every 360 degrees, which is another way of saying that if you
move 360 degrees around a circle, you end up where you started.
Anyway, this shape is what is normally thought of as a 'wave'.  Lots
of things in nature can be understood in terms of waves, so much so
that even though the history of the sin and cos functions is mostly
about triangles, the major uses of the functions today have to do with
describing waves.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Trigonometry

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