1. Toothpicks and PowerPoint Ambiguous Case of the Law of Sines Pam Burke Potosi High School #1 Trojan Drive Potosi, MO 63664 573-438-2156 pburke@potosir3.org pamburke74@gmail.com

2. To solve a right triangle, you can use the Pythagorean Theorem and/or basic trig ratios. C A B a b c

3. In an oblique triangle (one in which there are no right angles) neither of those methods will work. Solving an oblique triangle requires using either the Law of Sines or the Law of Cosines, depending on which information you are given.

4. In this lesson we are going to use the Law of Sines.

5. The Law of Sines can be used when you are given two angles and one side of an oblique triangle (ASA or AAS).

6. Here is an example. You are given two angles and one side of a triangle. Find the measures of the other angle and the other two sides. A B C a b c A = 65º a = 28 B = 32º b = ? C = ? c = ?

7. A B C a b c A = 65º a = 28 B = 32º b = ? C = ? c = ? C = 180º - (A + B) = 180º - (65 + 32) = 180º - 97º = 83º

8. A B C a b c A = 65º a = 28 B = 32º b = ? C = 83º c = ?

9. A B C a b c A = 65º a = 28 B = 32º b ≈ 16.4 C = 83º c = ?

10. A = 65º a = 28 B = 32º b ≈ 16.4 C = 83º c ≈ 30.7 A B C a b c Now you have solved the triangle; you know the measures of all three sides and all three angles.

11. The Law of Sines can also be used to solve an oblique triangle when you are given two sides and the angle opposite one of the sides (SSA).

12. b a A Suppose you are given the measures shown below for two sides and an angle of an oblique triangle.

13. b A a The pieces could be put together to form this triangle.

14. b A a But they could also be put together to form this triangle.

15. Since SSA does not always define a unique triangle, this is called the Ambiguous Case of the Law of Sines. b A a b A a