Today we will focus on solving for the sides and angles of non-right triangles when given two angles and a side.

Transcription

1 5.5 The Law of Sines: Part 1 Pre-Calculus Learning Targets: 1. Use the Law of Sines to solve non-right triangles. Today we will focus on solving for the sides and angles of non-right triangles when given two angles and a side. Derivation: The Law of Sines Example 1: In ABC, A 49, a = 3 and B 0. Solve ABC. Example : The bearings of two ramps on the shore from a boat are 115 and 13. Assume the two ramps are 855 feet apart. How far is the boat from the nearest ramp on shore if the shore is straight and runs north-south? 5 1

2 Example 3: US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? 140 Pelican Bay mi 1/8 mi 1/8 mi 135 US 41 Example 4: An adventurer who is stuck on the top of a cliff is trying to decide whether or not he can jump to the next ledge. In his free time he is able to determine the angle from the bottom of the tree to the edge of the cliff and the angle from the top of the tree to the edge of the cliff as shown in the diagram below. While he was climbing the tree to measure that second angle he figured out that the tree is 13 feet tall. Find the distance d the adventurer will have to jump in order to make it to the ledge (assuming he doesn t trip over the roots of the tree and fall to the bottom of the cliff). 38 d 15 Example 5: An emergency dispatcher must determine the position of a caller reporting a fire. Based on the caller s cell phone records, she is located in the area shown. Overcome by the desire to solve for any missing lengths, the dispatcher momentarily forgets about the fire and wants to know what the unknown side lengths are in the triangle. Tower 60.1 Tower mi Tower 3 5

3 5.6 The Law of Cosines Pre-Calculus Learning Targets: 1. Use the Law of Cosines to solve non-right triangles.. Find the area of a non-right triangle when given SAS. 3. Find the area of a non-right triangle when given SSS using Heron s Formula. The Law of Sines works well when we are given AAS, ASA or when we have a magic pair. But what if we are not able to find an angle and the side across from it? What if we are given or? The Law of Cosines: a b c bccosa Note that we must know two sides to use this law. We must also know either the angle A between the two sides (SAS) or the third side a. Once a problem is started with Law of Cosines, you SHOULD continue with this law to find all missing pieces! Example 1: Solve WXY if x = 17cm, y = 6. cm and W = Area of a Non-Right Triangle: Area ab sin C Example : Find the area of ABC when A 49, c = 13 and b =

4 While Area 1 ab sin C works to find the area when given SAS what if we are given SSS? Heron s Formula: When given a, b, and c in a non-right triangle 1 Semiperemeter : S a b c Area s s a s b s c Example 3: Bob wants to sod a portion of his backyard roughly in the shape of a triangle with sides 9 feet, 1 feet and 15 feet. How many 4.5 square foot sod rolls does Bob need to buy? 14. Example 4: An airplane flies north from Ft. Myers to Sarasota a distance of 150 miles, and then changes his bearing to 50 degrees and flies to Orlando, a distance of 100 miles. a) How far is it from Ft. Myers to Orlando? b) What bearing is needed for the ilot to return from Orlando to Ft. Myers? Example 5: Two planes that were flying together in formation take off in different directions. One plane goes East at 350 mph, and the other plane goes ENE at 380 mph. (The angle between E and ENE is.5j... you re welcome ). How far apart are the planes two hours after they separate? 5 4

5 5-5 The Law of Sines: Part - Ambiguous Case Pre Calculus Learning Targets: 1. Determine when a triangle is not possible using Geometry and the Law of Sines.. Determine when a situation yields two triangles and solve for BOTH triangles using the Law of Sines. So, what about SSA? We never used SSA in Geometry! Remember AAS, ASA, SAS, and SSS were theorems in Geometry because those pieces always created two congruent Triangles. Using the given pieces of a triangle (already drawn below) and a ruler we are going to create triangles. Notice one side is dashed because its length is unknown. AB = 4.5 cm A 30 BC = 3 cm B A The problem with Two Sides and a NON-Included Angle is there is more than one possibility for your answers! We call it the ambiguous case. No s: 1 : s: Example 1: Determine from the given information if zero, one or two triangles may exist. Explain and then solve for the remaining parts of the triangle. a) W = 56, w = 30, x = 6 c) A = 38, b = 1, a = 14 b) R = 15, r = 6, s =

6 c) A = 38, b = 1, a = 14 Example : Solve the triangle in Example 1c. Example 3: On Spring Break, Bob and his friend decide to go 4-wheeling off road in his new Jeep. The Jeep has a winch (a lifting device with a cable) that is used to pull the Jeep in case it gets stuck. After driving too fast over a hill, Bob finds himself stuck in the middle of a shallow stream. While wading through the stream to attach the cable to a tree a hill on the other side of the stream (see diagram), Bob ponders the mathematics of his situation He wonders what the angle of elevation of the cable was before his Jeep was pulled to the edge of the stream. 100 ft ft 5 6