The Law
of SINES
When Do I use
Law of Sines vs. Law of Cosine ?
Two sides
One opposite angle
given
given
Two angles
One opposite side
given
given
Two side
One angle
given
side
Given three
sides
any angle
side
given
Helpful Web Site

http://www.mathwarehouse.com/trigo
nometry/law-of-sines-andcosines.php
Use Law of SINES when ...
you have 3 dimensions of a triangle and you need to find the other
3 dimensions - they cannot be just ANY 3 dimensions though, or
you won’t have enough info to solve the Law of Sines equation.
Use the Law of Sines if you are given:

AAS - 2 angles and 1 adjacent side

ASA - 2 angles and their included side

SSA (this is an ambiguous case)
Example 1
You are given a triangle, ABC, with
angle A = 70°, angle B = 80° and side a
= 12 cm. Find the measures of angle C
and sides b and c.
* In this section, angles are named with capital
letters and the side opposite an angle is named
with the same lower case letter .*
Example 1 (con’t)
B
The angles in a ∆ total 180°,
so angle C = 30°.
80°
a = 12
c
A 70°
b
Set up the Law of Sines to
find side b:
C
12
b

sin 70 sin 80
12 sin 80  b sin 70
12 sin 80
b
 12.6cm
sin 70
Example 1 (con’t)
B
80°
c
A 70°
Set up the Law of Sines to
find side c:
a = 12
b = 12.6
30°
12
c

sin 70 sin 30
C
12 sin 30  c  sin 70
12 sin 30
c
 6.4cm
sin 70
Example 1 (solution)
Finally! Gott’em all
A 70°
B
Angle C = 30°
80°
Side b = 12.6 cm
a = 12
b = 12.6
30°
Side c = 6.4 cm
Note:
C
We used the given values of A
and a in both calculations. Your
answer is more accurate if you
do not used rounded values in
calculations.
Example 2
You are given a triangle, ABC, with
angle C = 115°, angle B = 30° and side
a = 30 cm. Find the measures of angle
A and sides b and c.
Example 2 (con’t)
To solve for the missing sides or
angles, we must have an angle and
opposite side to set up the first
equation.
B
30°
c
a = 30
115°
C
b
We MUST find angle A first because
the only side given is side a.
A
The angles in a ∆ total 180°, so angle
A = 35°.
Example 2 (con’t)
B
Set up the Law of Sines to find side b:
30
b

sin 35 sin 30
30°
c
a = 30
115° 35°
C
b
A
30 sin 30  b sin 35
30 sin 30
b
 26.2cm
sin 35
Example 2 (con’t)
B
Set up the Law of Sines to find side c:
30°
c
a = 30
115° 35°
C
b = 26.2 A
30
c

sin 35 sin 115
30 sin 115  c  sin 35
30 sin 115
c
 47.4cm
sin 35
Example 2 (solution)
done! Got all parts 
B
Angle A = 35°
30°
Side b = 26.2 cm
c = 47.4
a = 30
115° 35°
C
b = 26.2 A
Side c = 47.4 cm
Note: Use the Law of Sines
whenever you are given 2
angles and one side!
The Law of Sines
a
b
c


sin A sin B sin C
Use the Law of Sines to find
the missing dimensions of a
triangle when given any
combination of these
dimensions.

AAS
 ASA
Applying Law of Sines
Due next class worksheet problems
# 1