Review Solving a right triangle 2 Given two

Review Solving a right triangle. 2. Given two sides. 3. Given one angle and one side. 1.

6. 1: Law of Sines Objectives: n. Use the Law of Sines to solve oblique triangles n. Find areas of oblique triangles n. Use Law of Sines to model & solve real-life problems

Oblique Triangles n Oblique triangles do not have right angles. n Triangles are usually labeled as: C a b A c B

2 Types of Oblique Triangles All angles are acute. One angle is obtuse. C C b A a h h c B b a B A c

Must haves for solving Oblique Triangles n 2 angles and any side (AAS or ASA) n 2 sides and an angle opposite one of them (SSA) n 3 sides (SSS) n 2 sides and their included angle (SAS) n The first 2 cases can be solved using the Law of Sines n The last 2 cases can be solved using Law of Cosines

What is the Law of Sines? n Follow the directions on the Law of Sines Discovery notes ( available on mrtower. wordpress. com )

Law of Sines If ABC is a triangle with sides a, b, c, then: It can also be written as its reciprocal

2 Angles & 1 Side (AAS) Given: C=102. 3º, B=28. 7º, b=27. 4 feet Find: Finishing solving the triangle Label the givens. 2. Solve for the missing angle. 3. Use the Law of Sines to find the 2 missing sides. 4. A = 49 degrees, a = 43. 06 ft, c = 55. 75 ft 1. C a b A c B

2 Angles & 1 Side (ASA) Given: A pole tilts toward the sun at an 8º angle from the vertical, and it casts a 22 foot shadow. The angle of elevation from the tip of the shadow to the top of the pole is 43º. Find: How tall is the pole? C b a A c Label the givens. Solve for the missing angle. Use the Law of Sines to find the 2 missing sides. b = 23. 84 ft, B a = 34. 62 ft

The Ambiguous Case am·big·u·ous n Adjective n /amˈbigyo oəs/ – (of language) Open to more than one interpretation; having a double meaning. – Unclear or inexact because a choice between alternatives has not been made. n Synonyms – equivocal - vague - uncertain - doubtful - obscure

The Ambiguous Case (SSA) This one is a pain… in the SSA. Three possible situations: 1. No such triangle exists. 2. Only one such triangle exists. 3. Two distinct triangles can satisfy the conditions.

Example n Show that there is no triangle for which a=15, b=25, & A=85° 1. 2. b A a h 3. 4. 5. Label the givens & draw picture. Use the Law of Sines to find the missing angle B. Is this result valid? Why or why not? Invalid since out of Range sin. B = 1. 66

Example Given: triangle ABC where a=22 inches, b=12 inches, & A=42° Find: the remaining side and angles. C a b A Label the givens & draw picture. 2. Use the Law of Sines to find the missing angle B. 3. Solve for C. B 4. Solve for c. 1. c 5. B=21 o C=117 o c=29. 29 in

Example n Find 2 triangles for which a=12 meters, b=31 meters and A=20. 5° 1. 2. 3. 4. 5. 6. 7. Label the givens & draw both pictures. Use the Law of Sines to find the missing angle B 1. Subtract B 1 from 180° to find B 2 Subtract the B and A values from 180° to find C 1 and C 2. Use the Law of Sines to find c 1 and c 2. Solution 1: B=64. 8 o C = 94. 7 o c = 34. 15 m Solution 2: B=115. 2 o C=44. 3 o c= 23. 93 m

Area of an Oblique Triangle The area of an oblique triangle given some angle is half the product of the two adjacent sides and the sine of

Example n Find the area of a triangular lot having 2 sides of lengths 90 meters and 52 meters and an included angle of 102° Label the givens & draw picture. Use the Area Formula to find the area of the lot. Area = 2, 288. 82 m 2

Homework n Check Blackboard. n Check mr. Tower. wordpress. com for all notes, slides, and practice worksheets.