1. Chapter 6 Additional Topics in Trigonometry

2. 6.2 The Law of Cosines Objectives:  Use Law of Cosines to solve oblique triangles (SSS or SAS).  Use Law of Cosines to model & solve real-life problems.  Use Heron’s Area Formula to find areas of triangles. 2

3. Proof of Law of Cosines  Consider a triangle with three acute angles.  The coordinates of the vertices are A(0, 0), B(c, 0), and C(x, y).  The altitude of the triangle is y.  We see that x = b cos A and y = b sin A. 3

4. Proof continued….  Side a is the distance from B to C. 4

5. The Law of Cosines 5

6. Example 1  Find the angles of the triangle with sides a = 8 feet, b = 19 feet, and c = 14 feet. Hint: Find the largest angle first. 6

7. Why Find the Largest Angle First?  What do we know about the cosine of an angle in QI ? In QII ?  In a triangle:  If cos θ > 0, then θ is ____________.  If cos θ < 0, then θ is ____________. 7

8. Example 2  Find the remaining angles and side of a triangle given A = 115°, b = 15 cm., and c = 10 cm. 8

9. Example 3  The pitcher’s mound on a women’s softball field is 43 feet from home plate. The distance between the bases is 60 feet. (The pitcher’s mound is not halfway between home plate and 2 nd base.) How far is the pitcher’s mound from first base? 9

10. Heron’s Area Formula  Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by 10

11. Example 5  Find the area of a triangle having sides of lengths a = 43 m, b = 53 m, and c = 72 m. 11

12. Formulas for Area of Triangle  Right Triangle:  Oblique Triangle:  Heron’s Formula (given all 3 sides): 12

13. Homework 6.2  Worksheet 6.2 13