



Most people know kind of what an angle is.




You can see them at the point of a slice of pizza ...












The "lift" in a skateboard jump ramp ...












and stuff like that all have angles we know about.






To get really technical,



if you take two different rays that start at the same point



you get an angle ...












There are lots of ways to measure angles.



The way that gets used the most is called degrees.






Why do they call it degrees?



Who knows!



And who really cares?






They
could have called it anything, 


degrees
is just a name. 





The
important thing is how it works. 





It works
kind of like a crazy watch that runs backwards. 





Example: 


Start
with a ray at 3 O'clock ... 











Now put
another ray at 12 O'clock. 











The angle
measure from the 3 O'clock ray 


to the 12
O'clock ray ... 











Now take
away the ray at 12 O'clock 


and put
in one at 9 O'clock ... 











There are
3 hours from 12 O'clock to 3 O'clock 


and 90
degrees. 


There are
6 hours from 9 O'clock to 3 O'clock. 


Guess how
many degrees it is? 











Now take
away the ray at 9 O'clock 


and put
one in at 6 O'clock. 





It's 9
hours from 6 O'clock to 3 O'clock 


How many
degrees do you think it is? 











The degrees are the measurement of how far you must spin



(math types say rotate)



to get from pointing in the direction of one line to pointing 


in the direction of the other. 





The next thing you need to know is a bit about is triangles.






What makes a triangle a triangle?



Three straight lines all connected end to end to end. 


All three lines might be the same length, 


two might be the same length or none of the lines might be the same length. 





In geometry class, you probably had to learn all kinds of names 


like equilateral, and obtuse, and iscoceles, and stuff like that. 


Well, in trig we couldn't care less about those names and postulates 


and all those things. 





In fact, there's only 3 things we really need from geometry. 





1) If you add up the 3 angles on the inside of a triangle you get 180 degrees. 


No matter what the triangle looks like.






2) If one of the angles on the inside of the triangle is 90°, 


the triangle is called a right triangle. 


The triangle is drawn with a little square on the 90° angle 


to show that it is 90° 





The side of the triangle across from (called opposite) the 90° angle 


(labeled c here) is the longest side of the triangle. 


It gets a special name. It's called the hypotenuse. 


If
you multiply the hypotenuse times itself, 


the
answer you get is always the same number that you get 


when
you multiply the other two sides times themselves 


and
then add that together. 


That
little trick is called the Pythagorean Theorem. 





3) (The most important one.) 


Say we have a small right (one 90° angle) triangle. 





and we make a really big copy of it. 





now even though the copy is much bigger, it's still exactly the same shape. 


That means the angles are still the same. 


The pointy little angle on the right 


is the same number of degrees in the little triangle as it is in the big copy. 


The other angle (in the upper left) is still the same size, 


and the 90° angle is still 90°. 


OK, Big deal, so what? 


Here's the deal. 


Let's label the sides of the little triangle a, b, and c, 


and let's label the sides of the big copy A, B, and C. 








Suppose the ratio of the length of b to the length of c



(which is the fraction ^{ b}/c) is:






THEN ON THE BIG TRIANGLE:









That means, 


that if we have a right triangle of any size with these same angles, 


the length of the side across from the little pointy angle on the right (B) 


divided by he length of the hypotenuse equals
^{ 1}/3. 





In geometry, this idea is called the law of similar triangles. 


When it was first
discovered, average people said 


"Oh, that's sort of neat, I guess" but math types
said: 


"Wow! I see big time job security here. 


We can take that little puppy and build
an entire area of math out of it. 


We'll call it trigonometry, and with
all the shenanigans we can do with it, 


we'll all have jobs for
life!" And they did. 


All of trigonometry comes from the law of similar triangles. 





copyright 2008 Bruce
Kirkpatrick 
