



So trigonometry comes from the ratios of the lengths




of the sides of right triangles to each other. 


If we think about the triangle from the view of that 


"pointy little angle on the right" you might have met on 


the last page, we could label the sides: 








Now we want to find the ratios of each of these sides to the others 


(that is, one
side /another). 


There are six ways that they can be put together. 





opposite
side 

hypotenuse 



hypotenuse 

opposite
side 



adjacent
side 

hypotenuse 



hypotenuse 

adjacent
side 



opposite
side 

adjacent
side 



adjacent
side 

opposite
side 






Math types decided it took too much time and effort to say 


"the ratio of the opposite side to the hypotenuse" and stuff like that 


so they invented words to stand or them. 


Sine, cosine, tangent, cotangent, secant, and cosecant. 


So: 





Sine
= 
opposite
side 

Cosecant
= 
hypotenuse 



hypotenuse 

opposite
side 





Cosine
= 
adjacent
side 

Secant
= 
hypotenuse 



hypotenuse 

adjacent
side 





Tangent
= 
opposite
side 

Cotangent
= 
adjacent
side 



adjacent
side 

opposite
side 






Remembering these things takes some practice, 


but let's cut it down a bit. 





If you
look at the three on the right, 


you will see that they are the three on the left flipped
upside down. 


Now all you have to do is remember the ones on the left. 


A memory
trick that might help you is the letter sequence 


SOH  CAH 
TOA that is ... 





Sine
= Opposite/Hypotenuse
SOH 


Cosine
= Adjacent/Hypoyenuse
CAH 


Tangent
= Opposite/Adjacent
TOA 





The other thing to keep straight is: 


Is the sine turned upside down the secant, or
the cosecant? 





The way I remember this is, 


the names group together in
pairs so that one (and only one) 


of the two has "co" in
it, one of the main three (sine, cosine, and tangent) 


always
goes with one of the other three ... 


AND
... 


Tangent goes with
cotangent, 





So
sine, one of our main three needs to go with one of the other three. 


Cotangent
is already taken, so we have secant and cosecant left. 


So
which is it? 


One
of the two needs to have "co" in the name ... 





So
sine goes with cosecant. 





That
leaves us with cosine and secant. 


So
everything works out. 





These
names are totally lame. 


They
really don't mean anything so you need to use 


brute
force memory power to get them. 


It's
a bad deal, but that's how it is, sorry. 





The
good news, is that almost everything else in trig makes logical
sense. 





So
wonderful, let's look at a problem ... 





Example: 





If we have a right triangle with an angle of, say, 23°. 


The side opposite the angle divided by the hypotenuse 


(a ratio known as the sine) 


is the same number no matter what the size of the triangle is. 











So what is the sine of 23°? 


We have 3 ways to find out. 


1) Punch the numbers into a calculator 


2) Look it up in a table (found in the back of trig books) 


3) Work out a long calculation using something called a Taylor Series 


(Choose number 1, Choose number 1) 





I get, the sine of 23° = .390731128. 


What do you get? 





Normally this is written sin 23° = .390731128 


with the "e" left off the end of the word sine. 





It's
amazing, 


changing stuff like "the ratio of the opposite side to the hypotenuse" 


to one
short word wasn't good enough, 


now they're
shortening
that one short word! 





The
abbreviations are: 





Sine 

sin 

Cosecant 

csc 
Cosine 

cos 

Secant 

sec 
Tangent 

tan 

Cotangent 

cot 






Let's look at another way that these ratios work out. 


Here's a triangle: 











We've labeled the two angles (that aren't the 90° angle) A and B, 


and we've named
the side opposite angle A, "a" 


and the side opposite angle B, "b". 


The hypotenuse is labeled h. 


So: 








Do you see that: 


Sin
A = Cos
B 
Cos
A = Sin
B 
Tan
A = Cot
B 
Sec
A = Csc
B 
Csc
A = Sec
B 
Cot
A = Tan
B 






The next idea comes from the first thing we got from geometry: 





"the sum of the angles on the inside of a triangle is 180°" 





If the right triangle uses up 90° of the 180°
total 


then the other two angles
add up to the other 90 degrees. 





A + B = 90° so B = 90°
 A 


A
= 90°  B B = 90°  A 


We just said: 


Sin
A = Cos B AND B = 90°  A 





so by substituting "90°
 A" for B we get: 





Sin A =
Cos(90°  A) 





We could substitute in all over the place for B in the stuff like 


csc A = sec B that
we came up with before but you get the idea. 





Now maybe someday you would need to find the square of sin 30°. 


The problem
is that big shot math types will laugh at you 


if you write (sin 30°)^{2}. 





Powers of trig ratios have there own special notation. 


The exponent comes after the word but
before the number. 


It is written sin
^{2 } 30°. 


Who knows why, but it is. 





OK, one more thing to talk about
on this page. 


The Pythagorean theorem says



The sum of the squares of the two sides of a right triangle 


equals the square of the
hypotenuse" 








that is: 


a^{2} + b^{2} = h^{2} 





Movie fans might remember that in The Wizard of Oz, 


the scarecrow
says this at the end after the wizard gives him a diploma. 





So the Pythagorean
theorem is really part of geometry, 


but there is
something very similar in trig. 


It goes
... 





sin
^{2}A + cos ^{2}A = 1 





This is called an identity, you'll see more of them later. 


They are kind of like puzzles. 


You change pieces of them around
until they turn out the way you want. 





Want to see how we get this one? 





Sure you do. 


It won't hurt... 


much. 














The identity equation is: 


sin
^{2}A + cos ^{2}A = 1 





writing that as a's and b's and h's we get: 






+ 

=
1 






We have a
common denominator on the left side, so we can add the fractions: 






=
1 






Now clear the
denominator by multiplying both sides by h ^{2}. 














Simplify.
That is, clear out the mess ... 


a^{2} + b^{2} = h^{2} 


And
there you are! 


copyright 2008 Bruce
Kirkpatrick 
