The LAW of SINES - MathHands
Trigonometry Sec. 05 notes
MathHands.com M´ arquez
The LAW of SINES Main Idea The Law of Sines is similar to the law of cosines in that it is often helpful in solving non-right triangles. Roughly speaking, the Law of Sines says that the ratios of a sides to the sine of corresponding angles are all equal. Number of Triangles We take a moment here to emphasize the importance of solving all equations correctly, and completely. By doing so, we are sure to find out the correct number of triangle solutions. In general, when 3 of the items are given and 3 are missing, there will always be 0, 1, 2, or infinite many real-triangle solutions. To determine the total number of real triangles solutions for each specific problem one needs only to solve the equations correctly. Observe the following examples, as some of these contain two triangle solutions. Law of Sines: sin A a
A b
c
B
sin A a
sin B b
C
a
=
sin B b
=
sin C c
sin C c
Example: Solve C 5
7
65◦
B
c ◦
We first note that we know the ratio sin765 , and we know the side b = 5. Thus.. we can apply the law of sines to solve for the angle B. The following diagram shows the ration we know and the ratio we will compare it to. C 5
7
we
w k no
65◦ c
c
2007-2009 MathHands.com
B
math hands
pg. 1
Trigonometry Sec. 05 notes
MathHands.com M´ arquez sin B sin 65◦ = 7 5 sin 65◦ = sin B 5· 7 sin 65◦ sin B = 5 · 7 sin B ≈ 0.647
(applying the law of sines) (algebra) (now, we will use calc. to approximate) (now we need to solve correctly and completely)
B ≈ . . . , −220.3 , 40.3 , 139.7 , 400.3◦, . . . ◦
◦
◦
This is as far as the law of sines can help us.. but we can take it from here. Assuming B is an interior angle of triangle [in flatland.. Euclidean space] we can assume it must measure somewhere between 0 and 180◦ . The only such choice in the above list of candidates for B are B ≈ 40.3◦ Note that 139.7◦ + 65 is more than 180◦ Thus not a viable option for an interior angle.
74.7◦ 5
7
65◦
40.3◦
c
Then we just solve for c... again by using the law of sines. c 7 = ◦ sin 74.7 sin 65◦ 7 c= · sin 74.7◦ sin 65◦ c ≈ 7.45
Finally, the solution,
74.7◦ 5
7
65◦ 7.45
40.3◦
Example: Solve C 7
6
20◦
B
c
c
2007-2009 MathHands.com
math hands
pg. 2
Trigonometry Sec. 05 notes
MathHands.com M´ arquez ◦
We first note that we know the ratio sin620 , and we know the side b = 7. Thus.. we can apply the law of sines to solve for the angle B. The following diagram shows the ration we know and the ratio we will compare it to. C 7
6
now we k
20◦
B
c
sin B sin 20◦ = 6 7 sin 20◦ 7· = sin B 6 sin 20◦ sin B = 7 · 6 sin B ≈ 0.399
(applying the law of sines) (algebra) (now, we will use calc. to approximate) (now we need to solve correctly and completely)
B ≈ . . . , −203.5◦, 23.5◦, 156.5◦ , 383.5◦, . . .
This is as far as the law of sines can help us.. but we can take it from here. Assuming B is an interior angle of triangle [in flatland.. Euclidean space] we can assume it must measure somewhere between 0 and 180◦ . The only such choices in the above list of candidates for B are B ≈ 23.5◦ or B ≈ 156.5◦ Thus we have two possible triangle solutions: 3.5◦
136.5◦
20◦
OR
23.5◦
c
7
20◦
6
7
6
156.5◦
c
then we just solve for c in each case... For the first triangle: 6 c = ◦ sin 136.5 sin 20◦ 6 c= · sin 136.5◦ sin 20◦ c ≈ 12.08
c 6 = ◦ sin 3.5 sin 20◦ 6 c= · sin 3.5◦ sin 20◦ c ≈ 1.07
or
3.5◦
136.5◦
20◦ 12.08
c
2007-2009 MathHands.com
OR
23.5◦
7
20◦
math hands
1.07
6
7
6
156.5◦
pg. 3
Trigonometry Sec. 05 exercises
MathHands.com M´ arquez
The LAW of SINES 1. Solve C 5
7
55◦
B
c
2. Solve C 5
4
20◦
B
c
3. Solve C 5
8
40◦
B
c
4. Solve C 5
7
40◦
B
c
5. Solve C 5
3
70◦
B
c
6. Solve
c
2007-2009 MathHands.com
math hands
pg. 4
Trigonometry Sec. 05 exercises
MathHands.com M´ arquez C 8
7
51◦
B
c
7. Solve C 8
16
51◦
B
c
8. Solve C 8
4
51◦
B
c
9. Solve C 5
4
41◦
B
c
10. Solve C 7
4
24◦
B
c
11. PROVE the Law of SINES 12. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there are infinite many possible real triangle solutions. 13. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there is NO possible real triangle solution. 14. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there are exactly 5 possible real triangle solutions.
c
2007-2009 MathHands.com
math hands
pg. 5
MathHands.com M´ arquez
The LAW of SINES Main Idea The Law of Sines is similar to the law of cosines in that it is often helpful in solving non-right triangles. Roughly speaking, the Law of Sines says that the ratios of a sides to the sine of corresponding angles are all equal. Number of Triangles We take a moment here to emphasize the importance of solving all equations correctly, and completely. By doing so, we are sure to find out the correct number of triangle solutions. In general, when 3 of the items are given and 3 are missing, there will always be 0, 1, 2, or infinite many real-triangle solutions. To determine the total number of real triangles solutions for each specific problem one needs only to solve the equations correctly. Observe the following examples, as some of these contain two triangle solutions. Law of Sines: sin A a
A b
c
B
sin A a
sin B b
C
a
=
sin B b
=
sin C c
sin C c
Example: Solve C 5
7
65◦
B
c ◦
We first note that we know the ratio sin765 , and we know the side b = 5. Thus.. we can apply the law of sines to solve for the angle B. The following diagram shows the ration we know and the ratio we will compare it to. C 5
7
we
w k no
65◦ c
c
2007-2009 MathHands.com
B
math hands
pg. 1
Trigonometry Sec. 05 notes
MathHands.com M´ arquez sin B sin 65◦ = 7 5 sin 65◦ = sin B 5· 7 sin 65◦ sin B = 5 · 7 sin B ≈ 0.647
(applying the law of sines) (algebra) (now, we will use calc. to approximate) (now we need to solve correctly and completely)
B ≈ . . . , −220.3 , 40.3 , 139.7 , 400.3◦, . . . ◦
◦
◦
This is as far as the law of sines can help us.. but we can take it from here. Assuming B is an interior angle of triangle [in flatland.. Euclidean space] we can assume it must measure somewhere between 0 and 180◦ . The only such choice in the above list of candidates for B are B ≈ 40.3◦ Note that 139.7◦ + 65 is more than 180◦ Thus not a viable option for an interior angle.
74.7◦ 5
7
65◦
40.3◦
c
Then we just solve for c... again by using the law of sines. c 7 = ◦ sin 74.7 sin 65◦ 7 c= · sin 74.7◦ sin 65◦ c ≈ 7.45
Finally, the solution,
74.7◦ 5
7
65◦ 7.45
40.3◦
Example: Solve C 7
6
20◦
B
c
c
2007-2009 MathHands.com
math hands
pg. 2
Trigonometry Sec. 05 notes
MathHands.com M´ arquez ◦
We first note that we know the ratio sin620 , and we know the side b = 7. Thus.. we can apply the law of sines to solve for the angle B. The following diagram shows the ration we know and the ratio we will compare it to. C 7
6
now we k
20◦
B
c
sin B sin 20◦ = 6 7 sin 20◦ 7· = sin B 6 sin 20◦ sin B = 7 · 6 sin B ≈ 0.399
(applying the law of sines) (algebra) (now, we will use calc. to approximate) (now we need to solve correctly and completely)
B ≈ . . . , −203.5◦, 23.5◦, 156.5◦ , 383.5◦, . . .
This is as far as the law of sines can help us.. but we can take it from here. Assuming B is an interior angle of triangle [in flatland.. Euclidean space] we can assume it must measure somewhere between 0 and 180◦ . The only such choices in the above list of candidates for B are B ≈ 23.5◦ or B ≈ 156.5◦ Thus we have two possible triangle solutions: 3.5◦
136.5◦
20◦
OR
23.5◦
c
7
20◦
6
7
6
156.5◦
c
then we just solve for c in each case... For the first triangle: 6 c = ◦ sin 136.5 sin 20◦ 6 c= · sin 136.5◦ sin 20◦ c ≈ 12.08
c 6 = ◦ sin 3.5 sin 20◦ 6 c= · sin 3.5◦ sin 20◦ c ≈ 1.07
or
3.5◦
136.5◦
20◦ 12.08
c
2007-2009 MathHands.com
OR
23.5◦
7
20◦
math hands
1.07
6
7
6
156.5◦
pg. 3
Trigonometry Sec. 05 exercises
MathHands.com M´ arquez
The LAW of SINES 1. Solve C 5
7
55◦
B
c
2. Solve C 5
4
20◦
B
c
3. Solve C 5
8
40◦
B
c
4. Solve C 5
7
40◦
B
c
5. Solve C 5
3
70◦
B
c
6. Solve
c
2007-2009 MathHands.com
math hands
pg. 4
Trigonometry Sec. 05 exercises
MathHands.com M´ arquez C 8
7
51◦
B
c
7. Solve C 8
16
51◦
B
c
8. Solve C 8
4
51◦
B
c
9. Solve C 5
4
41◦
B
c
10. Solve C 7
4
24◦
B
c
11. PROVE the Law of SINES 12. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there are infinite many possible real triangle solutions. 13. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there is NO possible real triangle solution. 14. Give an example of a triangle where 3 of the quantities: a, b, c, A, B, C are given and there are exactly 5 possible real triangle solutions.
c
2007-2009 MathHands.com
math hands
pg. 5
