1. Law of Sines and Law of Cosines Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 An oblique triangle is a triangle that has no right angles. Definition: Oblique Triangles 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. C BA a b c

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The following cases are considered when solving oblique triangles. Solving Oblique Triangles 1.Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) A C c A B c a c b C c a c a B

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.) Definition: Law of Sines Law of Sines If ABC is an oblique triangle with sides a, b, and c, then Acute Triangle C BA b h c a C B A b h c a Obtuse Triangle

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Find the remaining angle and sides of the triangle. Example: Law of Sines - ASA Example (ASA): The third angle in the triangle is A = 180  – C – B = 180  – 10  – 60  = 110 . C B A b c 60  10  a = 4.5 ft Use the Law of Sines to find side b and c.

6. The Ambiguous Case (SSA) Law of Sines In earlier examples, you saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, then three possible situations can occur: (1) no such triangle exists, (2) one such triangle exists, or (3) two distinct triangles satisfy the conditions.

7. The Ambiguous Case (SSA)

8. Example 3 – Single-Solution Case—SSA For the triangle in Figure 6.5, a = 22 inches, b = 12 inches, and A = 42 . Find the remaining side and angles. Figure 6.5 One solution: a  b

9. Example 3 – Solution Check for the other “potential angle” C  180  – 21.41  = 158.59 (158.59 + 42 = 200.59 which is more than 180  so there is only one triangle.) Now you can determine that C  180  – 42  – 21.41  = 116.59  cont’d Multiply each side by b Substitute for A, a, and b. B is acute.

10. Example 3 – Solution Then the remaining side is given by cont’d Law of Sines Multiply each side by sin C. Substitute for a, A, and C. Simplify.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Use the Law of Sines to solve the triangle. A = 110 , a = 125 inches, b = 100 inches Example: Single Solution Case - SSA Example (SSA): C  180  – 110  – 48.74  C B A b = 100 in c a = 125 in 110  = 21.26  Since the “given angle is already obtuse, there will only be one triangle.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Use the Law of Sines to solve the triangle. A = 76 , a = 18 inches, b = 20 inches Example: No-Solution Case - SSA Example (SSA): There is no angle whose sine is 1.078. There is no triangle satisfying the given conditions. C A B b = 20 in a = 18 in 76 

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Use the Law of Sines to solve the triangle. A = 58 , a = 11.4 cm, b = 12.8 cm Example: Two-Solution Case - SSA Example (SSA): a = 11.4 cm C A B1B1 b = 12.8 cm c 58  Example continues. C  180  – 58  – 72.2  = 49.8  Check for the other “potential angle” C  180  – 72.2  = 107.8  (107.8  + 58  = 165.8  which is less than 180  so there are two triangles.)

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Use the Law of Sines to solve the second triangle. A = 58 , a = 11.4 cm, b = 12.8 cm Example: Two-Solution Case – SSA continued Example (SSA) continued: B 2  180  – 72.2  = 107.8  C A B2B2 b = 12.8 cm c a = 11.4 cm 58  72.2  10.3 cm 49.8  a = 11.4 cm C A B1B1 b = 12.8 cm c 58  C  180  – 58  – 107.8  = 14.2 

15. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.) Definition: Law of Cosines Law of Cosines Standard FormAlternative Form Law of Cosines

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Find the three angles of the triangle. Example: Law of Cosines - SSS Example: C BA 8 6 12 Find the angle opposite the longest side first. Law of Sines:

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Solve the triangle. Example: Law of Cosines - SAS Example: Law of Sines: C BA 6.2 75  9.5 Law of Cosines: