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Pre-Calculus

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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.

Proof of Law of Cosines

- Consider a triangle with threeacute 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= bcosAand y = b sin A.

Proof continued….

- Side a is the distance from B to C.

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.

Why Find the Largest Angle First?

- What do we know about the cosine of an angle in QI? In QII?
- In a triangle:
- If cosθ > 0, then θ is ____________.
- If cosθ < 0, then θ is ____________.

Example 2

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

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 2nd base.) How far is the pitcher’s mound from first base?

Heron’s Area Formula

- Given any triangle with sides of lengths a,b, and c, the area of the triangle is given by

Example 5

- Find the area of a triangle having sides of lengths a = 43 m, b = 53 m, and c = 72 m.

Formulas for Area of Triangle

- Right Triangle:
- Oblique Triangle:
- Heron’s Formula (given all 3 sides):

Homework 6.2

- Worksheet 6.2