Quiz 5-5 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. A = ? 2. a = ? 3. c = ? 4. Are there 0, 1, or 2 triangles for this case? A = 36.9º, b = 5, a = 4

HOMEWORK  Section 5-6 (page 494) (evens) 2, 6 -10, 14, 18, 20, 26-30, 38, 48 (12 problems)

5.6 The Law of Cosines

What you’ll learn about  Solving Triangles (SAS, SSS)  Triangle Area Applications … and why The Law of Cosines is an important extension of the Pythagorean theorem, with many applications.

Your turn: 1. Which of the cases above can be solved using the Law of Sines? 2. Which of the cases is the “ambiguous case” for the Law of Sines? AAA, AAS, ASA, SSS, SAS, SSA AAS, ASA, SSA SSA

A B a C There is a pattern for Law of Cosines c There are six possible unknowns in a triangle (3 sides, 3 angles). b A problem will give you three of the six unknowns. After labeling the triangle with the given information, draw the following pattern (loop the the given information, draw the following pattern (loop the 1 side and its opposite angle and the 2 other sides. 1 side and its opposite angle and the 2 other sides. If three of the four items circled are known, use law of cosines.

Law of Cosines

Triangle Review If the following information is given: C a “Walk around the block” Start at the first side or angle that is known then list the order of the known then list the order of the known items. known items. Side, Angle, side  SAS Law of Sines will not work for SAS but Law of Cosines will º B

SAS The “nice one” C a º B 1 st step: Law of Cosines: 2 nd step: either Law of Sines or Law of cosines. Law of cosines. For SAS: generally use Law of Sines to solve for one of the two remaining angles (because inverse sine function can distinguish between obtuse and acute angles. between obtuse and acute angles.

B a C There is a pattern for Law of Cosines If it is not already given, draw and label a triangle Draw the Law of Cosines pattern. (loop the 1 side and its opposite angle and the 2 other sides. If three of the four items circled are known, use law of cosines. 20º

B 6.5 C 11 5 Use Law of Sines to solve for one of the two remaining angles (because inverse sine function can distinguish inverse sine function can distinguish between obtuse and acute angles. between obtuse and acute angles.

B º 144.8º 15.2º

Your turn: 3. a = ? 4. C = ? 5. B = ? B a C º A

B 5.6 C 4 6 Use Law of Sines to solve for one of the 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 Triangle Review If the following information is given: C 7 9 “Walk around the block” Start at the first side or angle that is known then list the order of the known then list the order of the known items. known items. Side, side, side  SSS Law of Sines will not work for SSS but Law of Cosines will.

5SSS This is the “sticky second step” C st step: Law of Cosines (pick the biggest angle 1 st ) AND 2 nd Step: do Law of Cosines AGAIN!!!!!

B 9 C There is a pattern for Law of Cosines 7 5 If three of the four items circled are known, use law of cosines. 50.7º

A 9 Try Law of sines (for the 2 nd step) º

9 A Try Law of Cosines (for the 2 nd Step) º 2 nd step Law of Sines: 2 nd step Law of Cosines: Which one is correct?!!!

Your turn: 6. Solve the triangle. 23 C B A 33.5º

Your turn: 6. Solve the triangle. 23 C B A 33.5º SSS: This is the “silly second step”

C a “Walk around the block” Side, Angle, side  SAS Law of Sines will not work for SAS but Law of Cosines will º B A

C 6.5  SAS º B A

B 6.5 Draw the cosine pattern º Find the last angle. C = 180 – 20 – 14.8 B = 14.8º 14.8° C A C = °

Your turn: 7. c = ? 8. A = ? 9. B = ?

C “The shortest distance between two points is a straight line.” B A SSS: does it form a triangle? AB represents the distance between points A and B. “Going from “A” directly to “B” is shorter than going from “A” to “B” via “C”.

C B A SSS: does it form a triangle? AB = 1 1 BC = 2 AC = 3 23 check check NO!!! 3 = 3

Your turn: 10. AB = 4, AC = 6, BC = AB = 19, AC = 14, BC = 3

Summary SSS 1. Always solve for the largest angle first 2. Use Law of Cosines to find the second angle 3. Calculate third angle SAS 1. Use Law of cosines to find missing side. 2. Use Law of sines to find the second angle 3. Calculate third angle

Area of a Triangle C B 6.5 5A 144.5º Area = ½(5)(6.5)sin(144.5) Area = 9.4 Area = ½*b*c*sin A Area = ½*a*c*sin B Area = ½*a*b*sin C A = 144.5º, b = 5, c = 6.5

Your turn: C B 8 7A 25º Area = ½(7)(8)sin(25) Area = 11.8 Area = ½*b*c*sin A Area = ½*a*c*sin B Area = ½*a*b*sin C A = 25º, b = 7, c = Find the area of triangle ABC