



There are lots of formulas




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 


sin^{2}X + cos^{2}X = 1 





#5 


To get this one, just multiply both sides of #4 by
^{1}/sin^{2}X: 





sin^{2}X 
+ 
cos^{2}X 
= 
1 



sin^{2}X 
sin^{2}X 
sin^{2}X 






and simplify: 


1 + tan^{2}X
= csc^{2}X 





#6 


You probably see this one coming. 


To get the
next one, just multiply both sides of #4 by ^{1}/cos^{2}X: 





sin^{2}X 
+ 
cos^{2}X 
= 
1 



cos^{2}X 
cos^{2}X 
cos^{2}X 






and simplify: 





tan^{2}X + 1 = sec^{2}X 





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

