Sin and Cosine rules
Naming non rightangled triangles

 CAPITALS are used
for angles (usually clockwise)
 lower case letters
used for sides.
The name of a side is the lower case of the letter that
represents the angle opposite that side

The sine rule
When finding
an angle, make it the numerator

To use the sine rule for
anything, we need to know one complete angle & side pair such as B and
b and another angle or side 
When finding a
side, make it the numerator

Example of use
What is the length of side x?
1) Find all the data needed 
180  (80 + 30) = 70 
2) Use the rule! 
^{x}/_{sin80 }= ^{10}/_{sin70}
x = ^{10}/_{sin70} * sin80
x = 10.5 (3s.f.) 
3) Does the answer make sense?
Use these simple tests to make sure you
haven't gone wrong...
1 Triangle's angles
add to 180!
2 The largest side in
a triangle is always opposite the largest angle
3 No side can be
greater than the sum of the other two!
There is also an extension to the sine
rule that you should know about. ^{sinA}/_{a} = ^{sinB}/_{b}
= ^{sinC}/_{c }= 2r
'2r' stands for twice the radius of the
circum circle that touches all three corners of the triangle
The cosine rule
To use the sine rule you need
one complete angle & side pair such as B and
b and another angle or side. Sometimes this
will not be provided so you might need to use the cosine rule...

To use the cosine rule, we
need to know...
all three sides to find an angle
and two sides and
the included angle to find a side. The
'included angle' is the angle between the known sides 

What is
the length of side b?
1) Find all the data needed (using the
sine rule in this particular case)

^{sinA}/_{20} = ^{
sin30}/_{10} sinA = ^{sin30}/_{10}
* 20
sin A = 1
sin^{1}(1) = 90
A = 90
so B = 60 
2) Use the rule!

b^{2} = 10^{2}
+ 20^{2}  (2*20*10*cos60)
b^{2} = 300
b = √300
b = 17.3 (3s.f.) 
3) Does the answer make sense?
Use these simple tests to make sure you
haven't gone wrong...
1 Triangle's angles
add to 180!
2 The largest side in
a triangle is always opposite the largest angle
3 No side can be
greater than the sum of the other two!
Finding areas of triangles

There are two formulas for
the area of a triangle 

1) Find all the data needed (using the
sine rule in this particular case)

cosA = (6^{2 }+ 4^{2 }
 5.5^{2}) / (2*4*6) cosA = 0.453125
A = 63.1 (3sf) 
2) Use the rule! 
1/2 * 4 * 6 * sin63.1
= 10.7 (3sf) 
