Quiz 5-5 Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º...
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Quiz 5-5 Quiz 5-5 e for the missing angle and sides of for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º riangle ABC where B = 25º, b = 15, C = 107º A = ? = ? = ? = ? c = ? = ? Are there 0, 1, or 2 triangles for this cas Are there 0, 1, or 2 triangles for this cas A = 36.9 A = 36.9 º, b = 5, a = 4 º, b = 5, a = 4
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Transcript of Quiz 5-5 Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º...
Quiz 5-5Quiz 5-5Solve for the missing angle and sides of Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º Triangle ABC where B = 25º, b = 15, C = 107º 1. 1. A = ?A = ?2. 2. a = ? a = ? 3. 3. c = ?c = ?
4. 4. Are there 0, 1, or 2 triangles for this case?Are there 0, 1, or 2 triangles for this case?
A = 36.9A = 36.9º, b = 5, a = 4º, b = 5, a = 4
HOMEWORKHOMEWORK
Section 5-6 (page 494)Section 5-6 (page 494)
(evens) 2, 6 -10, 14, 18, (evens) 2, 6 -10, 14, 18, 20, 26-30, 38, 4820, 26-30, 38, 48
(12 problems)(12 problems)
5.65.6
The Law of CosinesThe Law of Cosines
What you’ll learn aboutWhat you’ll learn about Solving Triangles (SAS, SSS)Solving Triangles (SAS, SSS) Triangle Area ApplicationsTriangle Area Applications
… and why
The Law of Cosines is an important The Law of Cosines is an important extension of the Pythagorean theorem, extension of the Pythagorean theorem, with many applications.with many applications.
Your turn:Your turn:
1. 1. Which of the cases above can be solved using Which of the cases above can be solved using the Law of Sines?the Law of Sines?
2. 2. Which of the cases is the “ambiguous case” Which of the cases is the “ambiguous case” for the Law of Sines?for the Law of Sines?
AAA, AAS, ASA, SSS, SAS, SSAAAA, AAS, ASA, SSS, SAS, SSA
AAS, ASA, SSAAAS, ASA, SSA
SSASSA
AA BB
aa
CC
There is a There is a patternpattern for Law of Cosines for Law of Cosines
cc
There are There are sixsix possible unknowns in a possible unknowns in a triangle (3 sides, 3 angles).triangle (3 sides, 3 angles).
bb
A problem will give you A problem will give you threethree of of the the sixsix unknowns. unknowns.
After labeling the triangle withAfter labeling the triangle with the given information, draw the the given information, draw the following pattern (loop thefollowing pattern (loop the 1 side 1 side and its and its opposite angleopposite angle and and the the 2 other sides.2 other sides.
If If threethree of the four items circled of the four items circled are known, use are known, use law of cosines.law of cosines.
Law of CosinesLaw of Cosines
Cabbac
Baccab
Abccba
ABC
cos2
cos2
cos2
Then way.usual in the
labeled angles and sides with leany triang be Let
222
222
222
ab
bacC
ac
cabB
bc
cbaA
2cos
2cos
2cos
222
222
222
Triangle ReviewTriangle ReviewIf the following information is given:If the following information is given:
CC
aa
““Walk around the block”Walk around the block”
Start at the first side or angle that isStart at the first side or angle that is known then list the order of theknown then list the order of the known items.known items.
Side, Angle, sideSide, Angle, side SASSAS
Law of Sines will not work for Law of Sines will not work for SASSAS but but Law of Cosines will. Law of Cosines will.
55
1111
2020ººBB
SASSAS The “nice one”The “nice one”
CC
aa55
1111
2020ººBB
11stst step: Law of Cosines: step: Law of Cosines:
22ndnd step: either Law of Sines or step: either Law of Sines or Law of cosines.Law of cosines.
For SAS: generally use For SAS: generally use Law of Sines Law of Sines to solve for one of the two remaining to solve for one of the two remaining angles (because inverse sine angles (because inverse sine function can distinguish function can distinguish between obtuse and acute angles.between obtuse and acute angles.
BB
aa
CC
There is a There is a patternpattern for Law of Cosines for Law of Cosines
1111
1. If it is not already given, draw and 1. If it is not already given, draw and label a triangle.label a triangle.
55
2. Draw the Law of Cosines 2. Draw the Law of Cosines pattern. pattern.
(loop the (loop the 1 side 1 side and its and its opposite opposite angleangle and the and the 2 other sides.2 other sides.
If If threethree of the four items circled are known, use of the four items circled are known, use law of cosines.law of cosines.
Abccba cos2222
2020ºº
20cos)11)(5(2115 222 a
6.422 a
5.6a
BB
6.56.5
CC
1111
55
2020ºº5.6
20sin
5
sin
B
Use Use Law of Sines Law of Sines to solve for one of to solve for one of the two remaining angles (becausethe two remaining angles (because inverse sine function can distinguish inverse sine function can distinguish between obtuse and acute angles.between obtuse and acute angles.
5.6
20sin5sin 1B
3.15B
BB
6.56.5
1111
55
2020ºº
144.8144.8ºº 3.1520180 C
7.144C15.215.2ºº
Your turn:Your turn:.65 and ,4 ,6 given that Solve AcbABC
3. 3. a = ?a = ?
4. 4. C = ? C = ?
5. 5. B = ?B = ?
BB
aa
CC
44
66
6565ºº
AAAbccba cos2222
65cos)4)(6(246 222 a7.312 a6.5a
BB
5.65.6
CC
44
66
6565ºº
6.5
65sin
6
sin
B
Use Use Law of Sines Law of Sines to solve for one of to solve for one of the two remaining angles (becausethe two remaining angles (because inverse sine function can distinguish inverse sine function can distinguish between obtuse and acute angles.between obtuse and acute angles.
6.5
65sin6sin 1B
2.76B 2.7665180 C
8.38C
55
Triangle ReviewTriangle ReviewIf the following information is given:If the following information is given:
CC
77
99
““Walk around the block”Walk around the block”
Start at the first side or angle that isStart at the first side or angle that is known then list the order of theknown then list the order of the known items.known items.
Side, side, sideSide, side, side SSSSSS
Law of Sines will not work for Law of Sines will not work for SSSSSS but but Law of Cosines will. Law of Cosines will.
55
SSSSSS This is the “This is the “ssticky ticky ssecond econd sstep”tep”
CC
77
99
11stst step: Law of Cosines (pick the step: Law of Cosines (pick the biggest angle 1biggest angle 1stst))
ANDAND
22ndnd Step: Step: do Law of Cosines AGAINdo Law of Cosines AGAIN!!!!!!!!!!
BB
99
CC
There is a There is a patternpattern for Law of Cosines for Law of Cosines
77
55
If If threethree of the four items circled of the four items circled are known, use are known, use law of cosines.law of cosines.
Ccos)9)(5(2957 222
7.50C
)9)(5)(2(
812549cos 1C
Cabbac cos2222 50.750.7ºº
AA
99
Try Law of sines (for the 2Try Law of sines (for the 2ndnd step) step)
77
5550.750.7ºº
9
sin
7
7.50sin A
2.84A
7
7.50sin9sin 1A
99
AA
Try Law of Cosines (for the 2Try Law of Cosines (for the 2ndnd Step)Step)
77
55Acos)7)(5(2759 222
7.95A
)7)(5)(2(
492581cos 1C
Abccba cos2222
50.750.7ºº
2.84A22ndnd step Law of Sines: step Law of Sines:
22ndnd step Law of Cosines: step Law of Cosines:
Which one is correct?!!!Which one is correct?!!!
Your turn:Your turn:.14 and ,23 ,25 given that Solve cbaABC
6. 6. Solve the triangle.Solve the triangle.
2323
CC
1414
2525
BBAA
Cabbac cos2222
Ccos)23)(25(2232514 222
)23)(25)(2(
529625196cos 1C
5.33C
33.533.5ºº
Your turn:Your turn:.14 and ,23 ,25 given that Solve cbaABC
6. 6. Solve the triangle.Solve the triangle.
2323
CC
1414
2525
BBAA
Abccba cos2222
Acos)14)(23(2142325 222
)14)(23)(2(
196529625cos 1A
1.81A
33.533.5ºº
5.331.81180 B 4.65B
SSS: This is the “SSS: This is the “sillysilly ssecond econd sstep”tep”
CC
aa
““Walk around the block”Walk around the block”
Side, Angle, sideSide, Angle, side SASSAS
Law of Sines will not work for Law of Sines will not work for SASSAS but Law of Cosines will. but Law of Cosines will.
55
1111
2020ººBB
.11 and ,5 ,20 given that Solve cbAABC
Abccba cos2222 20cos)11)(5(2115 222 a
6.422 a
5.6a
AA
CC
6.56.5
SASSAS
55
1111
2020ººBB
.11 and ,5 ,20 given that Solve cbAABC
5.6a
AA
BB
6.56.5
Draw the cosine patternDraw the cosine pattern
1111
55
2020ºº
Find the last angle.Find the last angle.
C = 180 – 20 – 14.8 C = 180 – 20 – 14.8
B = 14.8B = 14.8ºº
Bcos)11)(5.6(2115.65 222
966783.0cos B
966783.0cos 1B
Baccab cos2222
)11)(5.6)(2(
115.65cos
222
B
14.8°14.8°
CC
AA
C = 145.2C = 145.2
145.2°145.2°
Your turn:Your turn:
.25 and ,4 ,10 given that Solve CbaABC
7. 7. c = ?c = ?
8. 8. A = ? A = ?
9. 9. B = ?B = ?
CC ““The shortest distance between The shortest distance between two points is a straight line.”two points is a straight line.”
BBAA
SSS: SSS: does it form a triangle?does it form a triangle?
ABAB represents the distance between points A and B. represents the distance between points A and B.
ACBCAB
““Going from “A” directly to “B” is shorter Going from “A” directly to “B” is shorter than going from “A” to “B” via “C”.than going from “A” to “B” via “C”.
ABBCAC ABACBC
CC
BBAA
SSS: SSS: does it form a triangle?does it form a triangle?
AB = 1AB = 1
ACBCAB
ABBCAC
ABACBC 11
BC = 2BC = 2 AC = 3AC = 3
2233 321 checkcheck
132 checkcheck
123 NO!!! 3 = 3NO!!! 3 = 3
Your turn:Your turn:
triangle?a Is ABC10. 10. AB = 4, AC = 6, BC = 11AB = 4, AC = 6, BC = 11
11. 11. AB = 19, AC = 14, BC = 3 AB = 19, AC = 14, BC = 3
SummarySummarySSSSSS 1.1. Always solve for the largest angle firstAlways solve for the largest angle first2.2. Use Law of Cosines to find the second Use Law of Cosines to find the second
angleangle3.3. Calculate third angleCalculate third angle
SASSAS 1.1. Use Law of cosines to find missing side.Use Law of cosines to find missing side.2.2. Use Law of sines to find the second angleUse Law of sines to find the second angle3.3. Calculate third angleCalculate third angle
Area of a TriangleArea of a Triangle
CCBB
6.56.555
AA
144.5144.5ºº
Area = Area = ½(5)(6.5)sin(144.5)½(5)(6.5)sin(144.5)
Area = 9.4Area = 9.4
Area = Area = ½*b*c*sin A½*b*c*sin A
Area = Area = ½*a*c*sin B½*a*c*sin B
Area = Area = ½*a*b*sin C½*a*b*sin C
A = 144.5A = 144.5º, b = 5, c = 6.5º, b = 5, c = 6.5
Your turn:Your turn:
CCBB
8877
AA
2525ºº
Area = Area = ½(7)(8)sin(25)½(7)(8)sin(25)
Area = 11.8Area = 11.8
Area = Area = ½*b*c*sin A½*b*c*sin A
Area = Area = ½*a*c*sin B½*a*c*sin B
Area = Area = ½*a*b*sin C½*a*b*sin C
A = 25A = 25º, b = 7, c = 8º, b = 7, c = 8
12. 12. Find the area of Find the area of triangle ABCtriangle ABC
