Law of Sines and Law of Cosines Objectives: Use the Law of Sines and Cosines to solve oblique triangles

Oblique Triangles Triangles that have no right angles. As standard notation, the angles of a triangle are labeled A, B, and C and their opposite sides are labeled a, b, and c. To solve an oblique triangle, you need to know the measure of at least one side and any two other measures of the triangle – either two sides, two angles, or one angle and one side.

Four cases 1. Two angles and any side 2. Two sides and an angle opposite one of them 3. Three sides 4. Two sides and their included angle The first two cases can be solved using the Law of Sines, whereas the last two cases require the Law of Cosines

Law of Sines

Solve the triangle 1.

Solve the triangle 2. Two tracking stations are on an east-west line 110 miles apart. A large forest fire is located on a bearing of from the western station and a bearing of from the eastern station. How far is the fire from the western station?

The Ambiguous Case (SSA) In Examples 1 and 2, you saw that two angles and one side determine a unique triangle. However, if two sides and one opposite angle are given, then three possible situations can occur: 1) no such triangle exists 2) one such triangle exists 3) two distinct triangle may satisfy the conditions

Solve the Triangle 3.

Solve the Triangle 4.

Solve the Triangle 5.

Area of an Oblique Triangle The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle.

EX: Find the area of the triangle 6.

Law of Cosines

Solve the Triangle 7.

Solve the Triangle 8.

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

Heron’s Area formula

EX: Find the area of the triangle 10.