1. Math III Accelerated Chapter 13 Trigonometric Ratios and Functions 1

2. Warm Up 13.5 1. Solve triangle ABC. 2. Solve triangle ABC with B = 41 °, C = 90 °, and c = 22. 2 b = 13 A B C a c 33 °

3. 13.5 Apply the Law of Sines  Objective:  Solve triangles that have no right angle. 3

4. The Law of Sines 4  Law of Sines In any triangle ABC, with sides a, b, and c, C B A a b c

5. Use of Law of Sines  Use Law of Sines when we know:  One side and two angles  SAA or ASA  Two sides and an angle opposite  SSA  This may lead to more than one solution.  Recall from geometry – SAA and ASA prove triangle congruence, but SSA doesn’t. 5

6. Example 1  Solve triangle ABC with A = 28 °, B = 102 °, and a = 8. 6

7. The Ambiguous Case - SSA 7  In the ambiguous (SSA) case, there are three possibilities: 1. There is no triangle with the given information. 2. There is exactly one triangle with the given information. 3. There are two triangles, one acute and one obtuse, with the given information.

8. The Ambiguous Case - SSA 8  Given two sides and an angle opposite one of them, several possibilities exist:  No solution - side too short to make a triangle  One solution - side equals altitude 20° 10 3.42 20° 10 1

9. The Ambiguous Case - SSA 9  Two possible triangles could result.  One unique solution - the opposite side is longer than adjacent side. Solving for A could give either an acute or obtuse angle! 20° 10 5 A A' 20° 10 13.42

10. Example 2  Solve triangle ABC with A = 94 °, a = 18, and c = 13. 10

11. Example 3  Solve triangle ABC with A = 77 °, a = 6.1, and b = 9. 11

12. Checkpoints 1, 2, & 3  Solve triangle ABC. 1. C = 14 °, B = 117 °, b = 21 2. A = 56 °, a = 24, b = 16 3. B = 122 °, b = 5, a = 8 12

13. Example 4  Solve triangle ABC with A = 30 °, a = 10, and b = 15. 13

14. Area of a Triangle 14  In any triangle ABC, with sides a, b, and c, the area is given by:  A = ½ ab sin C  A = ½ ac sin B  A = ½ bc sin A C B A a b c

15. Example 5  A piece of land is bordered by three roads as shown. Find the area of the land. 15 B A C 1.4 mi 2.3 mi 78.1 °

16. Checkpoints 4 & 5 4. Solve triangle ABC with A = 35 °, a = 11, and b = 14. 5. Suppose the side lengths in Example 5 are 4.6 miles and 2.8 miles. Find the area. 16

17. Homework 13.5  Practice 13.5 17