Calculator for finding the Area of a circle
   
 There are lots of formulas
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 that math types have "built" out of trig identities.
  
 Some of these are very useful.
  
 Some are lame.
  
 In ancient times (20 years ago),
 calculators with trig functions cost really big bucks.

 Not many people had them.

  

 That meant memorizing these formulas was important.

   

 Now that you can get a trig function calculator for about two bucks, 

 you can probably get away with just knowing the most important ones.
 

 That would be the first eight.

 

 You may, of course, have a teacher that is as old as I am 

 who wants you to go through the same pain as they did

 and memorize all of this ... er, stuff.

  

 If that's the case, all I can say is sorry.

  
 OK, now for the good news.
 The first eight - the important ones.
 You probably already know!
  
 #1
 Remember when we drew a graph of the sine function?
 It looked something like this:

  
 The sine of 45° is about 0.7071, the sine of 90° is 1.
 The sine of -45° is about - 0.7071 the sine of -90° is -1.
 
 The negative angles have the same sine values as the positive angles,
 but with a minus sign.
 In math talk:

sin(-X) = -sinX

 That's #1!
 
 Functions where this happens are called odd.
  
 You can see that with an odd function like the sine,
 you could put a pin through the graph at the (0,0) point,
 spin the graph 180°, and have the same graph line.
 
 
 #2
 Now let's look at the cosine graph.
 
 The cosine of 30° is about 0.866
 The cosine of 60° is 0.5
 
 The cosine of -30° is about 0.866
 The cosine of -60° is 0.5
 
 The negative angles give the exact same answers as the positive angles.
 
 In math talk:
cos(-X) = cosX
 
 Functions where this happens are called even.
 
 With an even function, the -X side of the graph
 is the mirror image of the +X side.
 
  
 #3
 The tangent graph looks like this:
 
 
 Look at it.
 Is it a pin the (0,0) and spin it 180° like the sine graph?
 Or is it a or mirror left and right like the cosine?
 
 It's a pin and spin.
 That means

tan(-X) = -tanX

  
 With this pin and spin or mirror strategy,
 you should be able to figure out the formulas for the other three.
 
 But they are not part of the eight.
  
 We will get to do the next three all at once though.
 They are actually all the same one rearranged with a bit of algebra.
 The first one we've seen a bunch of times.
 
 #4
sin2X + cos2X = 1
 
 #5
 To get this one, just multiply both sides of #4 by 1/sin2X:
 
sin2X + cos2X = 1



sin2X sin2X sin2X
 
 and simplify:

1 + tan2X = csc2X

 
 #6
 You probably see this one coming.
 To get the next one, just multiply both sides of #4 by 1/cos2X:
 
sin2X + cos2X = 1



cos2X cos2X cos2X
 
 and simplify:
 
tan2X + 1 = sec2X
 
 If you forget those last two,
 they wouldn't be too hard to work out on the spot eh?
 
 Home stretch!
  
 #7 & 8
 Remember our right triangle stuff?
 
 
 And the geometry rule: 
 "The sum of the angles on the inside of the triangle add up to 180°"?
 
 Now since the right angle "uses up" 90° of the 180°,
 the other two angles add up to 90°.
 
 That is:

A + B = 90°,     B = 90° - A,   and like that

 
 The other thing that makes this work is that in the right triangle,
 the sine of one of the angles is the same as the cosine of the other angle:
 
 That is:
sin A = cos B
 
 Just a few lines ago we said that another name for B was 90° - A
 
 So let's substitute:
sin A = cos B
sin A = cos (90° - A)
 

 And by the exact same idea:

cos A = sin B

cos A = sin (90° - A)

  
 Up to now we were just thinking about this as parts of a right triangle
 with angles of less than 90°.
 The thing is, AND THIS IS IMPORTANT,
 if it works with those angles it works with ANY angles anywhere we find them.
 
 That means it doesn't matter if the angle is tiny or gigantic,
 positive or negative.
 
 When you write that idea out in official math talk, you get:
 
 #7

sin A = ± cos (90° - A)

  
 #8

cos A = ± sin (90° - A)

 
copyright 2008 Bruce Kirkpatrick