



Up to now, we've only been talking about right triangles.




Trigonometry would be pretty lame 


if it didn't have ways of dealing with other triangles. 


Like, say, this one: 








OK, that "sum of the angles inside the triangle equals 180°" thing



works for any
triangle so we can find the measure of the other angle:






other angle =

180°  55°  45°

other angle =

80°







So now we know the angles, 


but we don't know the lengths of the other two sides. 


You might think this is the place we break out our new trig tricks. 


BUT WAIT! 


That new "sin = opposite over hypotenuse" stuff only works with right triangles. 


What do we do? 


We make some! 


Oh yeah, how? 


We just draw a line through the triangle like this: 








The line cuts the triangle into two pieces (both of which are triangles).



The line is drawn so that it makes two right angles



where it hits the side (at the bottom of the triangle).









So the triangle becomes two right triangles joined together. 


There's no big secret or trick to this. 


If you have a triangle, just draw a line from one corner to the opposite side 


and "say" that it makes 2 right angles. 


It works on any triangle. 


Sounds a
lot like cheating, but it's not. 


The only thing is that if the triangle has one angle of
more than 90°, 


the line must go from that angle to the opposite side. 





Back to the example. 


There are no angles larger than 90°, so choose any angle to draw the line from. 


I'll choose the 80° angle. 








Now just think of this as 2 different triangles glued together. 


The 80° angle at he top is now two different angles 


(that
just happen to add up to 80°). 


We can use the
"sum of the angles equals 180°" to find both of them: 





unknown angle 1 + 90° + 55° =

180°

unknown angle 1 =

180°  90°  55°

unknown angle 1 =

35°







unknown angle 2 + 90° + 45° =

180°

unknown angle2 =

180°  90°
 45°

unknown angle2 =

45°













See, the 2 angles at the top do add up to 80°! 


Now lets find some side lengths. 


We've got the length of the side on the left. Let's use it! 


First we'll find the length of he line we used to cut the triangle into two parts. 


In geometry they gave a line like his 2 names 


an "altitude" and a "perpendicular bisector" EGAD!!!) 





Now we're
in trig so we'll just call it "a". 








sin
55° =

a 



5 

5
x sin 55° = 
a 

(sin
55° = 0.8192) so: 


5
x 0.8192 = 
a 

4.096
= 
a 













Now lets find the length of the side on the right. 


We'll call it "b". 


sin 45° = 
4.096




b 



b
= 
4.096




sin 45° 

(sin 45°
= .7071) So:


b
= 
4.096




0.7071 

b
= 
5.793 












To find the length of the side on the bottom, 


we need to find the length of the two pieces it was divided into 


by the line we drew, and
then add them together. 





We can do it
with trig or Pythagoras. 


Let's do one of each. (We'll call them c and d) 





On the left: 


tan 35° =

c 



4.096 



4.096
x tan 35° = 
c 

(tan 35° = .7002)
So:



4.096
x 0.7002 = 
c 

2.8680
= 
c 










On the right: 


(4.096)
^{2} + d ^{2} = 
(5.7927)
^{2} 
16.777
+ d ^{2} = 
33.5554 
d
^{2} = 
33.5554
 16.777 
d
^{2} = 
16.777 
d
= 
4.096 






Isn't that interesting? 


"d" is the same length as that bisector line we drew! 


Can you figure out why? 





So the length of the side at the bottom is c + d or: 





side at bottom = 
2.868 + 4.096 
side at bottom = 
6.964 






So the original triangle is: 





The line down the middle and everything to do with it went away. 


It wasn't part f the triangle, 


it was just something we invented to help find the answers. 


If you don't already have one, 


you can probably see that getting a calculator with trig functions 


would save you a lot of work. 





Let's
try another one ... 





Example:









Find the length of the other side, and the other two angles. 





First, draw
" THE LINE" and call it "a" 








Now we can
calculate the length of "a" 


using the 30° angle and
the side of length 6. 





From the
30° angle, "a" is the side opposite. 


In the
right triangle on the left, the side of length 6 is the
hypotenuse. 


The trig
function that uses the side opposite and the hypotenuse is the
sine. 


So: 


Sin
30° = 
a 


6 



6
x Sin 30° = 
a 

(Sin
30° = 0.5) so: 


6
x 0.5 = 
a 

3
= 
a 










Now with Pythagoras and "angles equal 180°" we can do a bunch. 


Call the parts
o f the side on the bottom "c" and "d" again: 








Now go
get'em: 


3^{
2} + c^{ 2} = 
6^{ 2} 
3^{
2} + d^{ 2} = 
4^{ 2} 
9
+ c^{ 2} = 
36 
9
+ d^{ 2} = 
16 
c^{
2} = 
36  9 
d^{
2} = 
16  9 
c^{
2} = 
27 
d^{
2} = 
7 
c
= 
5.196 
d
= 
2.646 






So the
length of the side at the bottom is: 





c + d =
7.842 








Now we just need to find the
measure of the angle at the top 


and the angle at the bottom right. 


You can
find the measure of an angle in a right triangle using trig, 


if
you know the lengths of any two sides. 


For the
triangle on the left, we know the length of two sides to be 3
and 6. 


For the
triangle on the right, we know the length of two sides to be 3
and 4. 


Lets work
on the one on the right first. 


If we call the angle at the
bottom right "B", then: 








The side
of length 3 is opposite angle B. 


The side
of length 4 is the hypotenuse. 


The trig
function that uses the side opposite 


and the hypotenuse is the
sine. 


So: 








So what angle has
0 .75 for its sine? 


Remember,
"arcsin" one of the math codes for the question, 


"what angle has a sine of this?" 





arcsin 0.75 = ? 





The other
math code uses the 1 exponent like notation: 





sin^{
1} 0.75 = ? 





Either
way, you need to punch it into your calculator or look it up in
a trig table. 


The answer is: 





arcsin 0.75 = 48.59°

aka: 
sin^{
1} 0.75= 48.59° 












Now we could do a trig thing to get the angles at the top 


by finding the two parts and adding them together. 





The thing
is, we don't have to! 


We can do the much easier
"sum of the angles =180°" 


to get the entire top angle at
once. 





entire top angle + 30° + 48.59° = 
180° 
entire top angle = 
180°
 30°  48.59° 
entire top angle = 
101.41° 









The line through the triangle and everything to do with it went away again. 





Ya know, these things take a long time to figure out. 


I sure wish there were some short cuts we could use to get the answers. 





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QUIZ: 


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copyright 2008 Bruce
Kirkpatrick 
