Digital Lesson Law of Sines

Definition: Oblique Triangles An oblique triangle is a triangle that has no right angles. C B A a b c To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Oblique Triangles

Solving Oblique Triangles The following cases are considered when solving oblique triangles. Two angles and any side (AAS or ASA) A C c A B c 2. Two sides and an angle opposite one of them (SSA) C c a 3. Three sides (SSS) a c b c a B 4. Two sides and their included angle (SAS) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solving Oblique Triangles

Definition: Law of Sines The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Law of Sines If ABC is an oblique triangle with sides a, b, and c, then C B A b h c a C B A b h c a Acute Triangle Obtuse Triangle Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Law of Sines

Example: Law of Sines - ASA Example (ASA): Find the remaining angle and sides of the triangle. C B A b c 60 10 a = 4.5 ft The third angle in the triangle is A = 180 – A – B = 180 – 10 – 60 = 110. 4.15 ft 110 0.83 ft Use the Law of Sines to find side b and c. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Law of Sines - ASA

Example: Single Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 110, a = 125 inches, b = 100 inches C B A b = 100 in c a = 125 in 110 21.26 48.74 48.23 in C  180 – 110 – 48.74 = 21.26 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Single Solution Case - SSA

Example: No-Solution Case - SSA Example (SSA): Use the Law of Sines to solve the triangle. A = 76, a = 18 inches, b = 20 inches C A B b = 20 in a = 18 in 76 There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: No-Solution Case - SSA

Example: Two-Solution Case - SSA Example (SSA): a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the triangle. A = 58, a = 11.4 cm, b = 12.8 cm 49.8 72.2 10.3 cm C  180 – 58 – 72.2 = 49.8 Two different triangles can be formed. Example continues. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Two-Solution Case - SSA

Example: Two-Solution Case – SSA continued Example (SSA) continued: 72.2 10.3 cm 49.8 a = 11.4 cm C A B1 b = 12.8 cm c 58 Use the Law of Sines to solve the second triangle. A = 58, a = 11.4 cm, b = 12.8 cm B2  180 – 72.2 = 107.8  C  180 – 58 – 107.8 = 14.2 C A B2 b = 12.8 cm c a = 11.4 cm 58 14.2 107.8 3.3 cm Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Two-Solution Case – SSA continued

Area of an Oblique Triangle C B A b c a Find the area of the triangle. A = 74, b = 103 inches, c = 58 inches Example: 103 in 74 58 in Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Area of an Oblique Triangle

The flagpole is approximately 9.5 meters tall. Application: A flagpole at a right angle to the horizontal is located on a slope that makes an angle of 14 with the horizontal. The flagpole casts a 16-meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is 20. 20 A 70 Flagpole height: b 34 B 16 m C 14 The flagpole is approximately 9.5 meters tall. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Application