Calculator for finding the Area of a circle
   
So now that we know how these things work,
Related Chapters
   
we can graph them.
We use a special set of axis that has the degrees
of the angle where X usually goes,
and the trig function value where Y usually goes.
For the sine it looks like this:
 
 You can look up values in a table or use a calculator or whatever you like.
 You only need to find the values from  0° to 360° 
 because the numbers just keep repeating after that in both directions.
 
sin X° sin X° sin X° sin X°
10 0.1737 100 0.9848 190 - 0.1737 280 - 0.9848
20 0.3420 110 0.9397 200 - 0.3420 290 - 0.9397
30 0.5000 120 0.8660 210 - 0.5000 300 - 0.8660
40 0.6428 130 0.7660 220 - 0.6428 310 - 0.7660
50 0.7660 140 0.6428 230 - 0.7660 320 - 0.6428
60 0.8660 150 0.5000 240 - 0.8660 330 - 0.5000
70 0.9397 160 0.3420 250 - 0.9397 340 - 0.3420
80 0.9848 170 0.1737 260 - 0.9848 350 - 0.1737
90 1.0000 180 0.0000 270 - 1.0000 360 - 0.0000
 
 Graphing this, we get something like:
 
 
 When the angle gets to 360°, it is actually back to the place it started, 0°. 
 So for other areas of the graph, the pattern just repeats itself.
 
 
 If you look at the graph above,
 you might notice that when you replace X with -X,
 you get the same value for the sine but with the sign reversed + to - or - to +.
 
 Because of this, Sine is called an "odd" function.
 Math types mean that like even and odd, not like it's a weird function.
 Well even if it is, that's not what they meant.
 
 Now lets look at some values of the cosine function from 0° to 90°.
  
cos X°
0 1.0000
10 0.9848
20 0.9397
30 0.8660
40 0.7660
50 0.6428
60 0.5000
70 0.3420
80 0.1737
90 0.0000
 
Notice that the values of the cosine function from 0° to 90°
 are the same numbers as the values of the sine function from 90° to 180°.
That means you can think of the cosine function
 as the sine function moved 90° to the left on the graph.
This will work out just like translations in algebra functions,
 but we'll get to that a little later.
 
 For right now, let's graph the cosine function from 0° to 360°:

 Extending this out to a larger area we get:

For the cosine, if we replace X by -X we get the same cosine value.

 For example:

 

cos 180° = cos -180° = -1
 

Because we get the same answer when we replace X by -X,

 math types say the cosine is an even function.

 

Let's talk about the tangent function now.

Tangent is the sine divided by the cosine,

 so where the cosine is zero (90°, 270°, etc.)

 sin divided by cos is something with zero in the denominator.

 

 A fraction with zero in the denominator is undefined.

 That means the graph of the tangent will have undefined points at those values.
 From graphing in Algebra,
 you may remember that when a graph has an undefined point,
 there will either be a hole or an asymptote.
 
Since sine and cosine repeat, it is reasonable to expect tangent to repeat. 
It does.
Since sine and cosine repeat every 360°,
 we might expect tangent to repeat every 360° too.
IT DOESN'T!
 
Let's see what it does:
 
tan X° tan X° tan X° tan X°
10 0.1763 100 - 5.6713 190 0.1763 280 - 5.6713
20 0.3646 110 - 2.7475 200 0.3646 290 - 2.7475
30 0.5774 120 - 1.7321 210 0.5774 300 - 1.7321
40 0.8391 130 - 1.1918 220 0.8391 310 - 1.1918
50 1.1918 140 - 0.8391 230 1.1918 320 - 0.8391
60 1.7321 150 - 0.5774 240 1.7321 330 - 0.5774
70 2.7475 160 - 0.3646 250 2.7475 340 - 0.3646
80 5.6713 170 - 0.1763 260 5.6713 350 - 0.1763
90 undef 180 0.0000 270 undef 360    0.0000
 
  So the tangent function repeats every 180° from 0 to 360° it looks like:
 

 

 Extending this a ways in both directions we get:
 
 
 The tangent is equal to the sine divided by the cosine.
 The cotangent is just the tangent flipped over (math types say inverted).
 That is, the cotangent is the cosine divided by the sine.
 The graph of the cotangent will look a little like the graph of the tangent.
 But where the tangent equals zero, 
 the zero will be in the denominator of the cotangent making it undefined.
 The graph of the cotangent looks like this:
 

 
We have now looked at the graphs of four of the six trig functions. 
The last two are the secant and cosecant.
The secant is the inverse of the cosine. That is, 1 divided by the cosine. 
The cosecant is the inverse of the sine. That is, 1 divided by the sine.
We will graph the cosecant
 and leave the secant for you to graph on your own if you want.
 
 The sine function has a value of zero every 180°. 
 That means that the cosecant will have an undefined point 
 at those places since the zero will be in the denominator.
 
To graph the cosecant, we can graph the sine,
 and then just mirror it away from the X axis.
 Like this:
 

 
copyright 2008 Bruce Kirkpatrick