1 © 2011 Pearson Education, Inc. All rights reserved 1 © 2010 Pearson Education, Inc. All rights reserved © 2011 Pearson Education, Inc. All rights reserved Chapter 7 Applications of Trigonometric Functions

OBJECTIVES Area of a Triangle SECTION Find the area of SAS triangles. Find the area of AAS and ASA triangles. Find the area of SSS triangles. 3

3 © 2011 Pearson Education, Inc. All rights reserved AREA OF A TRIANGLE The area K of a triangle is or where b is the base and h is the height (the length of the altitude to the base).

4 © 2011 Pearson Education, Inc. All rights reserved AREA OF AN SAS TRIANGLE The area K of a triangle ABC with sides a, b, and c is In words, the area K of a triangle is one-half the product of two of its sides and the sine of the included angle.

5 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 1 Finding the Area of a Triangle Find the area of the triangle ABC in the figure. Solution K = bc sin A = (36)(29) sin 62° ≈ square feet

6 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding a Triangular Area Determined by Cellular Telephone Towers Three cell towers are set up on three mountain peaks. Suppose the lines of sight from tower A to towers B and C form an angle of 120º and the distances between tower A and towers B and C are 3.6 miles and 4.2 miles, respectively. Find the area of the triangle having these three towers as vertices.

7 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 2 Finding a Triangular Area Determined by Cellular Telephone Towers Find the area of a triangle with angle  = 120º included between sides of lengths b = 4.2 and c = 3.6. Solution = bc sin A = (4.2)(3.6) sin 120° = (4.2)(3.6) sin 60° = (4.2)(3.6) ≈ 6.55 square miles K

8 © 2011 Pearson Education, Inc. All rights reserved AREA OF AAS AND ASA TRIANGLES The area K of a triangle ABC with sides a, b, and c is

9 © 2011 Pearson Education, Inc. All rights reserved HERON ’ S FORMULA FOR SSS TRIANGLES The area K of a triangle ABC with sides of lengths a, b, and c is given by where is the semiperimeter.

10 © 2011 Pearson Education, Inc. All rights reserved EXAMPLE 4 Using Heron’s Formula Find the area of triangle ABC with a = 29 inches, b = 25 inches, and c = 40 inches. Round the answer to the nearest tenth. Solution First find s: Area