



To solve for the unknown angles or side lengths




in these triangles:






We need
another shortcut. 





The new short
cut looks kind of like the Pythagorean theorem. 


Actually, it
IS the Pythagorean theorem, 


modified to work with any triangle not
just right triangles. 





Here's how it
goes: 


For any
triangle 








The lengths of
the three sides, a, b, and c, and one angle, C, 


show up in this
formula. 


You need to
know three of the four things to find the fourth. 


You can set up
the labels any way you want on a triangle 


as long as an angle and
the side opposite that angle use the same letter. 


That means we
can write this formula as any of these: 











The point is, 


there is nothing special about any of the sides or angles in the
triangle. 


You can start with whatever sides and angle you like
depending 


on what missing info you are looking for at the moment. 





Example: 





Label it: 





Now solve for something, say angle C: 











Subtract 74 from each side: 











Divide each side by 70 and simplify: 








Now find
the angle that has a Cosine of 0.92857: 








Remember,
Cos^{1} means: "What angle has a cosine of" 





Filling
in the new thing we know we have: 








OK.
Now we have enough info to use the trick with the sines from the
last page, 


(the Law of
Sines), if we want. 


Most people
think that one is easier to use. 


For fun, let's use the new
trick again. 





Did
I say FUN! Yikes, I should get a life, eh? 





Here
we go. This time, let's find the measure of angle B. 


To do this,
use the version of the equation with the Cos B in it. 





Subtract
58 from each side: 








Divide
each side by 42 and simplify: 








Now
find the angle that has a Cosine of 0.78571: 





Filling
in the new thing we know we have: 





Now
we have one more thing to find. 


We could use the Law of Sines
from the last page 


or the new equation from this
page, 


but
we could also use the "Angles inside the triangle add up to
180°
" thing 


and it is WAY easier, so: 








Fill
in that and we're done: 











Example: 





Here's
one where we know the lengths of two sides 


and the measure of
the angle between them: 








We
can't use the Law of Sines from the last chapter on this one. 


Can
you see why? 


The
reason is that any pair of angle and side across has an unknown. 


That means when we write the equation it will have TWO unknowns. 


That's one more than we can deal with in an equation like that. 





Here
we go. Label the sides and angles: 





Choose
a form of the equation. 


We
have sides b and c and angle A so we need the version 


that
includes those three: 








Look up
and insert the Cosine of 35°
in the equation(0.8192) and carry on: 








Filling
in the new thing we know we have: 





Usually
in math when we find square roots, 


we need to think about both
the positive and negative roots. 


The thing with triangles is
that negative lengths don't really mean anything 


so we can
ignore the negative root here. 





Now
we have the measure of an angle and the length of the side
across from it. That means we COULD use the Law of Sines to find
the rest of the missing info. 








But
since the Law of Sines was LAST page, 


we will carry on with
our new formula. 





Let's
find the measure of angle C next. 





Since
we want to find angle C, 


we need the version of the equation
that has Cos C in it: 








Filling
in that, we have: 





Just
one more thing to find, the measure of angle B. 


Let's get lazy
and use the "Angles add up to 180°"
thing. 











Fill that
in and we're done: 





The
official name of the equation we have been using in this page 


SHOULD have been something to do with the Pythagorean Theorem. 


It isn't. 


Math types decided to call it THE LAW OF COSINES. 


I
guess since there was a cosine in the equation that made sense. 





It's
quite a stretch if you ask me. 





They
didn't. 





There is
one case where three pieces of information is just not enough. 


That
happens when we have the length of two sides 


and the measure of
an angle that is NOT between them. 


Like
this: 








The
reason we can't solve this one is that we can make 


two
completely different looking triangles 


that have the angle and
sides like we have here. 


We can make the one we have above, 


and
we can also make this one: 








The
way to look at this is that there are two different places 


that
the side with length of 6 can attach to make this triangle work
out 


with the numbers we have: 





Since
we can't tell which one of these it is from the information 


(and
you can't trust a picture on a math test), 


we can't solve this
one. 


It is called "The Ambiguous Case" 





Now I have
some more bad news. 


We haven't gone through all the trig stuff
we need to 


to show you how we know the Law of Cosines really
works. 


In other words, we can't do a PROOF yet even if we wanted
to. 





Yeah, I
know you're really broken up about that one. 





We will
get to it, eventually. 





The next
important thing we need to talk about 


is trig without triangles,
just angles. 





The place
to begin that, is to talk about graphing this stuff. 





copyright 2008 Bruce Kirkpatrick

