



So now that we know how these things work,




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. 





X° 
sin X° 

X° 
sin X° 

X° 
sin X° 

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°. 





X° 
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: 





X° 
tan
X° 

X° 
tan
X° 

X° 
tan
X° 

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

