6 1 Law of Sines Objectives To use
6. 1 Law of Sines Objectives: To use the Law of Sines to 1) solve oblique triangles 2) find the areas of oblique triangles 3) solve applied real-life problems
An oblique triangle is a triangle that has no right angles. C a b A c B 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. Need to know 3 measurements. 2
The following cases are considered when solving oblique triangles. 1. Two angles and any side (AAS or ASA) A A c c B C 2. Two sides and an angle opposite one of them (SSA) c C 3. Three sides (SSS) b a c a 4. Two sides and their included angle (SAS) c a B 3
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 h C a c Acute Triangle h B b a c A Obtuse Triangle B 4
Case 1 (ASA): Example 1. Use the Law of Sines to solve the triangle. C = 10 , a = 4. 5 ft, B = 60 C 10 The third angle in the triangle is A = 180 – C – B a = 4. 5 ft = 180 – 10 – 60 4. 15 ft b 60 = 110 B c A 0. 83 ft Use the Law of Sines to find side b and c. 5
Case 2 (SSA, Angle is obtuse): Example 2. Use the Law of Sines to solve the triangle. A = 110 , a = 125 inches, b = 100 inches C 21. 26 a = 125 in b = 100 in 110 A 48. 74 c B 48. 23 in C 180 – 110 – 48. 74 21. 26 6
Case 2 (SSA, Angle is obtuse): Example 3. Use the Law of Sines to solve the triangle. A = 114 , a = 85 inches, b = 94 inches b = 94 in A = 114° a = 85 in Conclusion: SSA, Angle is obtuse A is obtuse a>b One Solution a<b No Solution 7
Case 2. (SSA, Angle is acute): Example 4. Angle A is acute and a ≥ b One Solution a= 9 b=7 A = 74° 8
The Ambiguous Case (SSA, Angle is acute) Given: two sides and an angle opposite to one of the given sides Three possible situations : a b 1) no such triangle exists; A 2) one such triangle exits; 3) two distinct triangles may satisfy the conditions. 9
The Ambiguous Case (SSA, Angle is acute): A is acute a<b a > b • sin A Two Solutions a = b • sin A One Solution a b A a < b • sin A No Solutions 10
The Ambiguous Case (SSA, Angle is acute): Example 5. Use the Law of Sines to solve the triangle. A = 76 , a = 18 inches, b = 20 inches C b = 20 in a = 18 in 76 B A There is no angle whose sine is 1. 078. There is no triangle satisfying the given conditions. 11
The Ambiguous Case (SSA, Angle is acute): C Example 6. Use the Law of Sines to solve the triangle. A = 58 , a = 11. 4 cm, b = 12. 8 cm 49. 8 b = 12. 8 cm a = 11. 4 cm B 72. 2 58 c A 10. 3 cm C 180 – 58 – 72. 2 49. 8 Two different triangles can be formed. Example continues. 12
The Ambiguous Case (SSA, Angle is acute): Example 6. C C 1 =49. 8 Use the Law of Sines to solve the second triangle. A = 58 , a = 11. 4 cm, b = 12. 8 cm B 2 180 – 72. 2 107. 8 b = 12. 8 cm a = 11. 4 cm 72. 2 B 1 C 2 180 – 58 – 107. 8 14. 2 C c 1=10. 3 cm A C 2 =14. 2 a = 11. 4 cm B 1 58 b = 12. 8 cm 107. 8 58 A B 2 c 2 = 3. 3 cm 13
Summary: Number of Triangles Satisfying the Ambiguous Case Let sides a and b and angle A be given in triangle ABC. (The Law of Sines can be used to calculate sin B. ) 1. If sin B > 1, then no triangle satisfies the given conditions. 2. If sin B = 1, then one triangle satisfies the given conditions and B = 90°. 3. If 0 < sin B < 1, then either one or two triangles satisfy the given conditions a) If sin B = value, then let B 1 = sin-1 value, and use B 1 for B in the first triangle. b) Let B 2 = 180° – B 1. If A + B 2 < 180°, then a second triangle exists, and use B 2 for B in the second triangle. If A + B 2 > 180°, then there is only one triangle. 14
Area of an Oblique Triangle C Example 7. Find the area of the triangle. A = 74 , b = 103 inches, c = 58 inches 103 in a b A 74 c B 58 in 15
Example 8. Finding the Area of a Triangular Lot Find the area of a triangular lot containing sides that measure 24 yards and 18 yards and form an angle of 80° A = ½(18)(24)sin 80° A 212. 7 square yards 16
Example 9, 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. How tall is the flagpole? A 20 70 Flagpole height: b 14 34 C B 16 m The flagpole is approximately 9. 5 meters tall. 17
- Slides: 17