The Law of Sines

1
The Lawof Sines
2
What if the triangle we want to solve is NOT a
right triangle? In the next two sections well
develop ways of solving these triangles if we
know at least one side and two other pieces of
info (sides or angles or one of each).
?
c
a
?
?
b
Well label these triangles with sides a, b and c
and use their Greek alphabet counterparts for the
angles opposite those sides as shown above.
3
Draw a perpendicular line and call the length h.
We do this so that we have a right triangle which
we already know how to work with.
?
c
a
h
?
?
b
Lets write some trig functions we know from the
right triangles formed.
Solve these for h
Since these both h we can substitute
divide both sides by ac
ac
ac
4
This process can be repeated dropping a
perpendicular from a different vertex of the
triangle. What we get when we combine these is
What this says is that you can set up the ratio
of the sine of any angle in a triangle and the
side opposite it and it will equal the ratio of
the sine of any other angle and the side opposite
it. If you know three of these pieces of
information, you can then solve for the fourth.
5
There are three possible configurations that will
enable us to use the Law of Sines. They are
shown below.
You dont have an angle and side opposite it here
but can easily find the angle opposite the side
you know since the sum of the angles in a
triangle must be 180.
ASA
SAA
You may have an angle, a side and then another
angle
You may have a side and then an angle and then
another angle
What this means is that you need to already know
an angle and a side opposite it (and one other
side or angle) to use the Law of Sines.
SSA
You may have an angle, a side and then another
side
6
Solve a triangle where ? 55, ? 82 and c 9
Draw a picture (just draw and label a triangle.
Don't worry about having lengths and angles look
right size)
?
9
c
55
a
6.20
This is SAA
?
?
82
43
Do we know an angle and side opposite it? If so
the Law of Sines will help us determine the other
sides.
b
7.44
How can you find ??
Hint The sum of all the angles in a triangle is
180.
How can you find a? (Remember it is NOT a right
triangle so Pythagorean theorem will not work).
You can use the Law of Sines again.
7
Solve a triangle where ? 15, a 35 and c 5
15
5
Do we know an angle and side opposite
it? If so the Law of Sines will help us
determine the other sides.
This is ASA
35
How can you find ??
Hint The sum of all the angles in a triangle is
180.
How can you find b? (Remember it is NOT a right
triangle so Pythagorean theorem will not work).
You can use the Law of Sines again.