Demana, Waits, Foley, Kennedy 5.5 The Law of Sines

What you’ll learn about Deriving the Law of Sines Solving Triangles (AAS, ASA) The Ambiguous Case (SSA) Applications … and why The Law of Sines is a powerful extension of the triangle congruence theorems of Euclidean geometry.

Review  

Overview A triangle can be defined as long as we have three of the six parts and one of the parts is a side. 1) ASA or SAA 2) SSA (two sides and an angle opposite) 3) SAS (two sides and an included angle) 4) SSS Situations 1 & 2 can be solved using Law of Sines Situations 3 & 4 can be solved using Law of Cosines

Area of any triangle The area of a triangle with sides of lengths a and b and with an included angle θ is; A = 1/2 ab sinθ   3 10

Create your own Create as many problems as possible from the figure shown:

Law of Sines ***Angles and sides opposite use same letter***

Use area to prove Law of Sines

Example: Solving a Triangle Given Two Angles and a Side

Solution

Solution

A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angeles, 340 miles apart. At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be 60 degrees at Phoenix and 75 degrees at Los Angeles. How far is the satellite from Los Angeles?

Example: Solving a Triangle Given Two Sides and an Angle (The Ambiguous Case)

Solve the triangle

Solve triangle ABC, where angle A = 42 degrees and sides a & b are 70 mm & 122 mm respectively

Visual of Ambiguous Case

5.5 HW, Page 439 Be able to do 1 – 22; 27 - 35

Solution

Solution

Solution

Solution

Solution

Solution

Example: Finding the Height of a Pole 15ft 15º 65º B A C

Solution x 15ft 15º 65º B A C