Calculator for finding the Area of a circle
 If you punch sin 45° into your calculator,
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 it probably comes back with 0.70710678 
 ( Maybe a few more or less decimal places) 
 If you punch sin 135° into your calculator 
 you get the exact same number 0.70710678!
 To answer that question, we have to draw a right triangle
 and drop an X Y coordinate grid ( better known as a graph) on it. 
 Ouch! I hope the triangle wasn't hurt.
 Now the thing that happens, is that the side of the triangle at the bottom
 goes from the origin (0,0 point) on the left, to some X value on the right. 
 We'll call that X value: X (How original). 
 That means that the triangle side on the bottom 
 goes from the Joint (0, 0) to the point (X,0). 
 Also the side on the right goes from that point (X,0) 
 to a point with the same X value, and some Y value,
 we'll call the point (X, Y).


 The length of the side on the bottom is X,
 and the length of the side on the right is Y.
 OK, The last side of the triangle is the hypotenuse.
 For reasons that will make sense in a little while,
 we are going to call the length of the hypotenuse 1.
 So we have:


 Now let's look at the pointy little angle on the left.
 What's the sine of this angle (call it "A").
Sine A = Y = Y

 What is the Cosine of A?
Cosine A = X = X

 So because the hypotenuse is 1, 
 the length of the side on the right of the triangle 
 is the value of the cosine of the pointy little angle on the left. 
 And because the hypotenuse is 1, 
 the length of the side on the bottom of the triangle 
 is the value of the sine of the pointy little angle on the left.
 This is a very special way to arrange a triangle.
 Since the hypotenuse is equal to 1 
 the length of the triangle side running in the X direction 
  is always equal to the sine of the angle at the origin of the graph.
 In this arrangement, 
 the cosine of the angle at the origin is always 
 equal to the length of the triangle running in the Y direction.
 This arrangement is so special that math types call it "STANDARD POSITION"
 It follows directly from this 
 that the tangent of the angle at the origin 
 is equal to the sine (Y) divided by the cosine (X)
 You can think of the hypotenuse like the hour hand on a watch.
 At two o'clock, you have this:
 Since there are 90° from straight up to straight out to the right, 
 each hour on a 12 hour clock face is equal to 30°. 
 We are looking at the angle from the X axis (3 O'clock) 
 to one hour up at 2 o'clock where the hypotenuse is. 
 That makes the angle 30°
 We can say that angle A = 30°.
 We can look up the sine of 30°.
 It is 0.5
 We know that the sine of an angle is equal to
 the side opposite divided by the hypotenuse
 and the hypotenuse in this case is 1. 
 That means:
Sine 30° = Y = Y



0.5 = Y


 This is just what we expected.
 We know that the Cosine of an angle 
 is equal to the side adjacent divided by the hypotenuse 
 and in this case the hypotenuse is 1.
 That means ...
Cosine 30° = X = X

0.866 = X
 What we have done is say ... 
 If we think about a triangle where the hypotenuse is 1, 
 we really don't need the triangle at all. 
 We can have sines, cosines, and all of the rest. 
 We can go back and check it on a triangle if we want,
 but we really don't have to.
 Those might sound like a trip into "WHO CARES LAND," 
 but most uses of uses of trig do not deal with triangles. 
 This is the path that lets us step away from the triangles where trig was born 
 to the places where we have many other uses for it.
 Let's do a couple more of these just to make sure it makes sense.
 Let's try 1 O'clock:
 Two hours is equal to 60° so:
 The sine 60° equals 0.866 so the Y length on our triangle is 0.866
 The cosine of 60° equals 0.5 so the X length on our triangle is 0.5
 The tangent of 60° equals our Y length divided by our X length
 so the tangent of 60° equals 1.732
 Now a tricky one. Let's try 12 o'clock:
 12 O'clock means three hours from the X axis,
 and at 30° per hour that means 90°.
 OK, We might have a problem.
 At 12 o'clock we don't have a triangle.
 The length of the X side is zero and the Y side is the same as the hypotenuse,
 That's true, but the answer is: 
 Who cares!

 We were just using the triangles as a helper. Just look at the X and Y values.

 What's the value of X?  ZERO
 So if the Cosine of an angle is equal to X, What's the cosine of 90°?  ZERO!
 So what's the sine of 90°?
 Since at 12 O'clock the hour hand runs along the Y axis,
 the Y length is 1. That means: 

 The sine of 90° is 1 

 That one was a little tricky.
 The next one is the real doorway to the world of no triangles.
 Let's try 11 O'clock:

 Which if we keep on doing things the way we have,
 is an angle of 120° (4 hours back from 3 O'clock at 30° per hour)

 The problem starts when we make the triangle from this. 
 To get back from the hypotenuse to the X axis 
 and then back from there to the origin (0,0) point,
 the triangle we get is:

 Which has nothing to do with a 120° angle.
 This is the moment we've all been waiting for!
 We'll OK, this is the moment that I have been waiting for
 and you have been humoring me about.
 Let's compare the 60° angle (1 O'clock) and the 120° angle (11 O'clock)

 And punch up some values on a calculator:
Angle: 60°   Angle: 120°
Sine = Y = 0.866   Sine = Y = 0.866
Cosine = X = 0.5   Cosine = X = -0.5
Tangent = Y/X = 1.732   Tangent = Y/X = -1.732
 Except for some minus signs here and there, they're the same!
 Now let's compare 240° and 300°:

 And the values for these are:
Angle: 240°   Angle: 300°
Sine = Y = - 0.866   Sine = Y = - 0.866
Cosine = X = - 0.5   Cosine = X = 0.5
Tangent = Y/X = 1.732   Tangent = Y/X = -1.732
 Again we get the same numbers
 except sometimes the sign of the number is different.
 Let's take a look at what goes on here.
 The sine is the value of Y. At small angles, Y is small.

 As the angle gets bigger towards 90°, Y gets bigger too.

 When we get to 90° the triangle disappears, Y is equal to 1,
 so the sine of 90° is equal to 1.

 As the angle gets bigger than 90° 
 Y, and so also the sine, get smaller than 1. 

 As the angle gets to 180° Y, and so also the sine, go to 0.

 As the angle gets to be more than180° 
 Y, and so also the sine, become negative.

 The angle grows to be 270°, where the triangle becomes a line again.
 Y is the same length as the hypotenuse again,
 but this time it is going in the negative Y direction.
 So the sine of 270° is -1.

 After 270°, Y is still negative, but it is getting shorter.
 That is, it's a smaller negative number.

 When we get to 360°, Y has shrunk back to zero, so the sine of 360° is 0.

 360° is really the same place as 0°, so the sine of 0° is also 0.
 Just because we went all the way around the circle
 does not mean we have to stop.
 We can just keep going around and around.

 405°, for example, is the same place as 45°, just once more around the circle.
 That means the values we get for all the trig functions
 (Sine, Cosine, Tangent, etc) 
 will be the same for 405° as they were for 45°. 
 In fact, and here's the big idea, 
 if you start at any angle and add any number of 360°'s to it
 the trig ratios will all be the same.
 One last point. 
 Angles measures in the counter clockwise direction (like we have been doing) 
 are considered positive. 
 Angles measured in the clockwise direction are considered negative.
 That means that an angle like 120° is the same angle as -240° 
 AND will have all the same values of it's trig functions.

copyright 2008 Bruce Kirkpatrick