1. 10-2: The Law of Sines
©2008 Roy L. Gover (www.mrgover.com)
Learning Goals
•Derive the Law of Sines
•Solve triangles using the
law of sines

2. Important Idea
•The Law of Cosines can
be used to solve oblique
triangles when SAS or SSS
information is given.
•The Law of Sines provides
a way to solve triangles
when the Law of Cosines
does not work.

3. Analysis
A B
C
D
a
c
b 2r
Observations
BCD is a
rt. triangle
1.
3.
2
=
a
sinD
r
4. =
a
sin A
2r
∠ ≅ ∠2. A D
Case I

4. Analysis
A B
C
D
a
c
b
2r
Observations
3. =
b
sinD
2r
4. =
b
sinB
2r
CAD is a
rt. triangle
1.
∠ ≅ ∠2. B D
Case II

5. Analysis
A B
C
D
a
c
b
2r
Observations
2.∠ D C∠≅
3. =
c
sinD
2r
4. =
c
sin C
2r
ABD is a
rt. triangle
1.
Case III

6. Analysis
= ⇒ =
a a
sin A 2r
2r sin A
= ⇒ =
b b
sinB 2r
2r sinB
= ⇒ =
c c
sin C 2r
2r sin C

7. therefore:
= =
a b c
sin A sinB sin C
is the Law of Sines
It’s a beautiful thing!!!

8. Example
82°
200 ft
Find the
distance
across
the lake.
44°
=
° °
d 200
sin 82 sin54
54°
d

9. Important Idea
The Greek astronomer,
Hipparchus of Nicaea, used
the idea in the last example
called triangulation to
estimate the distance from
the earth to the moon in 150
B.C.

10. Try This
Solve
ABC if
A=32.2°,
B=57.7°
and
c=14.3
A B
C
C=90.1°,a=7.6
& b=12.1

11. Try This
Solve
ABC
if A=29.2°
B=62.3°
and
c=11.5
A B
C
C=88.5°,a=5.6
& b=10.2

12. Important Idea
To use the law
of sines, you
must know at
least one angle
and the measure
of the side
opposite that
angle.
A
a

13. Try This
Solve
ABC
if A=30°
b=5.6 and
c=11.5
A B
C
Can’t Solve

14. Summary
The following is true about
the law of sines problems we
have worked:
•two angles are given
•the side opposite one of
these angles is given

15. Important Idea
When one angle and two
sides (with one side across
from the angle) are given, the
following may be true:
•No triangle exists
•one triangle exists
•two triangles exist

16. Example 1
Solve
ABC if
A=60°
b=50 and
a=33
A B
C
60°
50
33
a<bsinA⇒no solution
50sin60°

17. Solve
ABC if
A=60°
b=50 and
a=45
A B
C
60°
50
45
Solution #1
Example 2

18. Solve
ABC if
A=60°
b=50 and
a=45
A B
C
60°
50 45
Solution #2
Example 2 (cont.)

19. Solve
ABC if
A=60°
b=50 and
a=45
A B
C
60°
50 45
bsina
bsina<a<b⇒2 solutions
Example 2 (cont.)

20. Example 3
Solve
ABC if
A=60°
b=50 and
a=65
A B
C
60°
50
bsin
a 65
Solution#1

21. Solve
ABC if
A=60°
b=50 and
a=65
A
C
60°
50
bsin
a 65
No solution#2
Example 3 (cont.)

22. Solve
ABC if
A=60°
b=50 and
a=65
A B
C
60°
50
bsin
a 65
a>bsina & a>b⇒1 solution
Example 3 (cont.)

23. Deep
Thoughts
Life and
mathematics are
sometimes
complicated

24. Lesson Close
What is the purpose of the
law of sines?