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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, 

     C  /  |
       /   |  A
      /    |

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:

                            /  |
          /|          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:

          / |      
     C  /   |      
       /    |  A   
      /     |      

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 

  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   

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 

  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 

  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

   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
      /     |      

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)
            /     |      

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 

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

               /  |      
           C  /   |  A
             /    |  
            /     |      

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? 

                / |      
               /  |      
        ladder/   |  10 feet
             /    |  
            /     |      

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

                / |      
               /  |      
        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? 

                / |      
               /  |      
              /   |  ?
             /    |  
            /     |      
             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. 

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Ian, The Math Forum   

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: 

  Frequently Asked Questions About Trigonometry - J. David Eisenberg   

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.


- Doctor Dyno, The Math Forum   

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:

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

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,   

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

      a=0    90    180  270    360

       |     y 
       |  y     y                         y = sin(a)
       |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. 

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Ian, The Math Forum   
Associated Topics:
High School Trigonometry

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