Calculator for finding the Area of a circle
 So trigonometry comes from the ratios of the lengths
Related Chapters
 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. 



Sine =  

opposite side 

Cosecant =  


hypotenuse opposite side 

Cosine =  

adjacent side

Secant =  


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 ...
 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. 


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

Cot B = 

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: 

 a2 + b2 = h2
 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 2A + cos 2A = 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... 



Sin A = 

 so Sin 2A = 


Cos A =

so Cos 2A = 


The identity equation is:

sin 2A + cos 2A = 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 ...

 a2 + b2 = h2

And there you are!

copyright 2008 Bruce Kirkpatrick