6.1 Law of Sines

Objectives Use the Law of Sines to solve oblique triangles. Find Areas of triangles. Use the Law of Sines to solve problems. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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

Definition: Law of Sines If ABC is an oblique triangle with sides a, b, and c, then C B A b h c a Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Law of Sines

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 – C – B = 180 – 10 – 60 = 110. 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

A Pain in the Angle Side Side (SSA) Let’s consider the case where we have an angle, an adjacent side, and an opposite side. For example, we have angle A, side b, and side a. Sometimes a is too short to reach. This triangle has no solution. A b a Sometimes a reaches and creates a triangle in only one way. This triangle has one solution. A b a

No triangle on this side b a No triangle on this side Sometimes a is so long it only reaches one way that does not create a triangle. This triangle has no solution. Sometimes a is just the right length so that it forms 2 different triangles. This triangle has 2 solutions. a A b

SSA (The Ambiguous Case) If a solution is found, always check for a possible second solution. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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: Single Solution Case - SSA Example (SSA): Is a 2nd solution possible? 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: Two-Solution Case - SSA Example (SSA): a = 11.4 cm C A B 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 Can 2 different triangles 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

Solve the triangle: C = 31, c = 29 cm, b = 46 cm You try: Solve the triangle: C = 31, c = 29 cm, b = 46 cm Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Area of an Oblique Triangle Area of aTriangle C B A b h c a Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Area of an Oblique Triangle

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

Area of an Oblique Triangle Find the area of a triangular lot having two sides of lengths 90 meters and 52 meters and an included angle of 102. You Try: 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 flagpole to the sun is 20. How tall is the flagpole? 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

Example The course for a race starts at point A and proceeds in the direction to point B, then in the direction to point C, and finally back to A. Point C lies 8 km. directly south of point A. Approximate the total distance of the race course. 19.453 km Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located N55°E from Able; the call to Station Baker indicates that the ship is located S60°E from Baker. How far is the ship from each station? 135.58 miles from Station Baker, 143.33 miles from Station Able

Example You are standing 40 meters from the base of a tree that is leaning 8° from vertical away from you. The angle of elevation from your feet to the top of the tree is . How tall is the tree? 16.2 meters

Example A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to the lighthouse is and 15 minutes later the bearing is . The lighthouse is located at the shoreline. Find the distance from the boat to the shoreline.

Plan of action Use the Law of Sines to find the hypotenuse of the blue right triangle and right triangle trig to find the distance to shore (d). 3.2 miles

The Leaning Tower of Pisa was originally 184. 5 feet high The Leaning Tower of Pisa was originally 184.5 feet high. After walking 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be 60 degrees. Find the angle at which the tower is leaning. About 5.3 degrees Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Homework Pg. 398 9-33 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved.