



Now maybe we know what sines and cosines and stuff like that ARE.







What do we do with them? 


We'll start out with a couple of easy problems and go from there. 





Example: 





In the triangle above, 


we know the length of one of the sides (4) 


and the measure if two of the angles 


(remember, the little square means the angle is 90°). 


Can we
figure out the length of the other two sides, 


and the measure of the third angle? 





The angle is easy, and we don't even need trig. 


We have a 90° angle and a 60°
angle. 


Since all of
the angles add up to 180°: 





180°  90° 
60° = 30° 





So one piece of the puzzle is in place. 





For the next step we do need trig. 


The sine of an angle is
equal to the opposite side of the triangle 


divided by the hypotenuse (S
 O  H), 


So: 





sin 30° = 







solving this for "h"
we get 


h = 




"sin 30" is just a number. 


We can use a calculator or a trig table to find it. 


sin 30 =
0.5, so we have: 


h = 







Now we
have: 





Almost done! The last thing is to find the length of the third side. 


We could use trig. 


We could use any one of these: 





Cos 30° = 
Adjacent 
or 
Sin 60° = 
Opposite 
or 
Tan 60° = 
Opposite 



8 
8 
4 






Any of these will give you the right answer if you don't make a mistake. 


We
could also use our old friend the Pythagorean theorem. 





a^{2} + b^{2} = h^{2} 


(or call h^{2},
c^{2} if you want) 





So : 


4^{2
}+ b^{2 }= 
8^{2} 
16
+ b^{2 }= 
64 
b^{2
}= 
64  16 
b^{2
}= 
48 
b
= 







which
you could simplify to: 





b
= 

b
= 




Some people might
say that this isn't any simpler than the square root of 48, 


but some instructors like it better if you write it that way. 


Some will will
want you to use a calculator
and get a decimal answer (6.9282). 


Hey, whatever makes them happy. 





Let's
try another one ... 





Example: 








Find the angles and the length of the other side. 


First, use Pythagoras to find the length of the
side: 





6^{2}+b^{2} = 
15^{2} 
36+b^{2} = 
225 
b^{2} = 
225  36 
b^{2} = 
189 
b = 

b = 

b = 

b = 
13.748 






It almost never works out so that all three sides are whole numbers. 


Now we
need to find the angles. 


It's time for trig. 


Let's call the pointy little angle on the right A", so: 





SinA = 6/15 = 0.4 





If we're using trig tables, find 0.4 in the table and see what angle has
0.4 as it's sine. 





If you're using a calculator, you MIGHT press these keys: 





I said MIGHT because many calculators have their own special way of doing
things. 





Anyway, the angle that has .4 as its sine is about 23.578°. 





Now we use the
"three angles inside a triangle add up to 180°" thing 


to
find the other angle: 





90°
+ 23.578° + the other angle = 
180° 
the
other angle = 
180°  90°
 23.578° 
the
other angle = 
66.422° 









MORE ABOUT
NOTATION: 


During that
problem, we used the phrase "the angle that has.4 as its
sine". 


That's
mouthful ! 


Math types
just couldn't leave that one alone. 


They had to
come up with some word or symbols to mean that. So they did. 


Twice! 


First they
came up with the word "arc" 


which they tacked onto the
front of sine, tangent and the others 


to get arcsine, arctan and so
on. 





So arcsin 0.4
means: "the angle that has 0.4 as its sine" 


Well after a
while, they decided that they didn't like that notation any more, 


so
they changed it. 





Now they use:
sin^{1} 0.4 


It still
means: "the angle that has 0.4 as its sine" 


But wait! That
looks like an exponent! And it's where the exponent should go! 


All true. 


Personally, 


I
didn't see anything wrong with the old "arctan" type
notation, 


but then, nobody asked me. 





copyright 2008 Bruce
Kirkpatrick 
