10-2: The Law of Sines
©2008 Roy L. Gover (www.mrgover.com)
Learning Goals
•Derive the Law of Sines
•Solve triangles using...
Important Idea
•The Law of Cosines can
be used to solve oblique
triangles when SAS or SSS
information is given.
•The Law o...
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
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
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
Analysis
= ⇒ =
a a
sin A 2r
2r sin A
= ⇒ =
b b
sinB 2r
2r sinB
= ⇒ =
c c
sin C 2r
2r sin C
therefore:
= =
a b c
sin A sinB sin C
is the Law of Sines
It’s a beautiful thing!!!
Example
82°
200 ft
Find the
distance
across
the lake.
44°
=
° °
d 200
sin 82 sin54
54°
d
Important Idea
The Greek astronomer,
Hipparchus of Nicaea, used
the idea in the last example
called triangulation to
estim...
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
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
Important Idea
To use the law
of sines, you
must know at
least one angle
and the measure
of the side
opposite that
angle.
...
Try This
Solve
ABC
if A=30°
b=5.6 and
c=11.5
A B
C
Can’t Solve
Summary
The following is true about
the law of sines problems we
have worked:
•two angles are given
•the side opposite one...
Important Idea
When one angle and two
sides (with one side across
from the angle) are given, the
following may be true:
•N...
Example 1
Solve
ABC if
A=60°
b=50 and
a=33
A B
C
60°
50
33
a<bsinA⇒no solution
50sin60°
Solve
ABC if
A=60°
b=50 and
a=45
A B
C
60°
50
45
Solution #1
Example 2
Solve
ABC if
A=60°
b=50 and
a=45
A B
C
60°
50 45
Solution #2
Example 2 (cont.)
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.)
Example 3
Solve
ABC if
A=60°
b=50 and
a=65
A B
C
60°
50
bsin
a 65
Solution#1
Solve
ABC if
A=60°
b=50 and
a=65
A
C
60°
50
bsin
a 65
No solution#2
Example 3 (cont.)
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.)
Deep
Thoughts
Life and
mathematics are
sometimes
complicated
Lesson Close
What is the purpose of the
law of sines?
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Hprec10 2

  1. 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. 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. 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. 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. 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. 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. 7. therefore: = = a b c sin A sinB sin C is the Law of Sines It’s a beautiful thing!!!
  8. 8. Example 82° 200 ft Find the distance across the lake. 44° = ° ° d 200 sin 82 sin54 54° d
  9. 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. 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. 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. 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. 13. Try This Solve ABC if A=30° b=5.6 and c=11.5 A B C Can’t Solve
  14. 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. 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. 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. 17. Solve ABC if A=60° b=50 and a=45 A B C 60° 50 45 Solution #1 Example 2
  18. 18. Solve ABC if A=60° b=50 and a=45 A B C 60° 50 45 Solution #2 Example 2 (cont.)
  19. 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. 20. Example 3 Solve ABC if A=60° b=50 and a=65 A B C 60° 50 bsin a 65 Solution#1
  21. 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. 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. 23. Deep Thoughts Life and mathematics are sometimes complicated
  24. 24. Lesson Close What is the purpose of the law of sines?

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