Proof of law of sines
a
b
c


sin A sin B sin C
The possible cases are
(1) ABC is acute (all angles are acute)
(2) ABC is obtuse (one obtuse angle)
(3) ABC is right
Within cases (2) and (3), the choice of which angle is obtuse or
right is arbitrary; it's not going to make difference if A  90 , B  90 , or C  90 as the
triangle can be oriented and labeled any way.
Case 1:
ABC is acute ( A  90, B  90, C  90 )
1. Orient the triangle any way you like (I'll put vertex B at the
top), and construct an altitude, dividing side length b into
b1 and b2 .
2. By definition, sin A  [Fill in 1]
3. By definition, sinC  [Fill in 2]
4. Therefore the ratio
sin A
 [Fill in 3]
sin C
5. And you can do a bit of algebra to get the proportions to appear as it does in the law of
sines.
[Do algebra.]
6. Since all angles are acute, this proof holds no matter how we orient the triangle, and
a
b
c


we may conclude
.
sin A sin B sin C
Case 2:
ABC is obtuse (say A  90, B  90, C  90 )
Suppose
ABC is obtuse, and we'll arbitrarily say that
B is the obtuse angle.
Orient the triangle so that vertex B is at the top,
and construct an altitude, dividing side length
b into b1 and b2 .
Since A and C are both acute, the proof that
a
c

is identical to that in Case 1, and there's no need to go through it again.
sin A sin C
b
What we need to do is bring
into the proportion, and that takes a slightly different
sin B
approach, since B is obtuse.
Orient the triangle so it sits on base AB , and construct an
altitude from vertex C . This requires you to extend
AB as shown.
1. Since B is obtuse, we define its sine in terms of its
supplement; by definition sin B  [Fill in 1].
2. However, sin(180  B) = [Fill in 2], and therefore,
sin B  [Fill in 2 again].
3. Also, sin A  [Fill in 3].
4. Write the appropriate proportion and rearrange until you get
[Do this]
The law of sines is now established for this case.
a
b

.
sin A sin B
Case 3:
ABC is right (say A  90, B  90, C  90 )
a
c

is identical to case 1 and 2 (orient B at the top). The new
sin A sin C
thing now is to bring in the right angle at B .
The part with
Orient as shown. No need to construct an altitude, since this is already
a right triangle.
1. By definition, sin A  [Fill in 1].
2. Since B  90 , by definition, sin B  [Fill in 2].
3. So sin A can be written as a ratio sin A 
sin A sin A

[Fill in 3].
1
?
4. Equate, shuffle, and come out with the correct proportion.
[Do it.]
And...done.