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

6.1: Law of Sines Objectives: Use the Law of Sines to solve oblique triangles Find areas of oblique triangles Use Law of Sines to model & solve real-life problems

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

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

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

What is the Law of Sines? 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 A B C a b c 1. 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 = ft, c = ft

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? A B C ab c Label the givens. Solve for the missing angle. Use the Law of Sines to find the 2 missing sides. b = ft, a = 34.62ft

The Ambiguous Case am·big·u·ous /am ˈ bigyooəs/ Adjective –(of language) Open to more than one interpretation; having a double meaning. –Unclear or inexact because a choice between alternatives has not been made. 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 Show that there is no triangle for which a=15, b=25, & A=85° A a b h 1. Label the givens & draw picture. 2. Use the Law of Sines to find the missing angle B. 3. Is this result valid? Why or why not? 4. Invalid since out of Range 5. sinB = 1.66

Example Given: triangle ABC where a=22 inches, b=12 inches, & A=42° Find: the remaining side and angles. A B C b a c 1. Label the givens & draw picture. 2. Use the Law of Sines to find the missing angle B. 3. Solve for C. 4. Solve for c. 5. B=21 o C=117 o c=29.29in

Example Find 2 triangles for which a=12 meters, b=31 meters and A=20.5° 1. Label the givens & draw both pictures. 2. Use the Law of Sines to find the missing angle B Subtract B 1 from 180° to find B 2 4. Subtract the B and A values from 180° to find C 1 and C Use the Law of Sines to find c 1 and c Solution 1: B=64.8 o C = 94.7 o c = 34.15m 7. Solution 2: B=115.2 o C=44.3 o c= 23.93m

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 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 = 2,288.82m 2

Homework Check Blackboard. Check mrTower.wordpress.com for all notes, slides, and practice worksheets.