L AW OF S INES AND L AW OF C OSINES Objectives: Use the Law of Sines and Cosines to solve oblique triangles Find the areas of oblique triangles
O BLIQUE T RIANGLES 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.
F OUR CASES 1.Two angles and any side (AAS or ASA) 2.Two sides and an angle opposite one of them (SSA) 3.Three sides (SSS) 4.Two sides and their included angle (SAS) The first two cases can be solved using the Law of Sines, whereas the last two cases require the Law of Cosines
L AW OF S INES
S OLVE THE TRIANGLE 1.
S OLVE 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?
T HE A MBIGUOUS C ASE (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
S OLVE THE T RIANGLE 3.
S OLVE THE T RIANGLE 4.
S OLVE THE T RIANGLE 5.
A REA OF AN O BLIQUE T RIANGLE The area of any triangle is one-half the product of the lengths of two sides times the sine of their included angle.
EX: F IND THE AREA OF THE TRIANGLE 6.
L AW OF C OSINES
S OLVE THE T RIANGLE 7.
S OLVE THE T RIANGLE 8.
S OLVE THE T RIANGLE 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?
H ERON ’ S A REA FORMULA
EX: F IND THE AREA OF THE TRIANGLE 10.