Unit 4: Trigonometry Minds On

Unit 4: Trigonometry Learning Goal: I can solve word problems using Sine Law while considering the possibility of the Ambiguous Case. Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case Given information What can be Found Law Required 1.Two angles and any side (AAS or ASA) 2.Two sides and the contained angle (SAS) 3.Three Sides (SSS) 4.Two sides and an angle opposite one of them (SSA) For Non-Right Triangles

Unit 4: Trigonometry Ambiguous: "Unclear or inexact because a choice between alternatives has not been made.“ The ambiguous case arises when we use Sine Law for SSA. Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case Whenever you want to use Sine Law to determine an angle when you are given SSA, you need to consider the ambiguous case. If you are given two angles (AAS, or ASA) you do not need to worry about ambiguity.

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case With SSA there are three possibilities: No triangle exists Only one triangle exists Two triangles exist

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case When the given angle in SSA is acute these are the following possibilities:

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case 1. No triangle exists If the “swinging arm” of the triangle is less than the height of the triangle, then no solution exists. With the swinging arm being shorter than the height it will never reach the base of the triangle.

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case 2. One triangle exists If the “swinging arm” of the triangle is equal to the height of the triangle, then only one solution exists. With the swinging arm being equal to the height it will only reach the base of the triangle once (when it is perpendicular to the base) Also, when the swinging side is greater than or equal to the other known side, there is also only one solution.

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case 3. Two triangles exists If the “swinging arm” of the triangle is greater than the height of the triangle, then two solutions exists. With the swinging arm being greater than the height it will reach the base of the triangle twice. Once making an acute angle with the base, and once making an obtuse angle with the base.

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case When the given angle in SSA is obtuse these are the following possibilities:

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case 1. No triangle exists If the “swinging arm” of the triangle is less than or equal to the other given side of the triangle, then no solution exists.

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case 2. One triangle exists If the “swinging arm” of the triangle is greater than the other given side of the triangle, then one solution exists.

Unit 4: Trigonometry Determine angle C Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Determine angle C Lesson 6 – Word Problems and the Ambiguous Case

A B 30° a = 10 c = 16 Unit 4: Trigonometry Determine angle C Lesson 6 – Word Problems and the Ambiguous Case A B C 30° a = 10 c = 16

Unit 4: Trigonometry What are angles of depression and angles of elevation? Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Lesson 6 – Word Problems and the Ambiguous Case

Unit 4: Trigonometry Lesson 5 – Word Problems Homework  Pg. 318 #1, 3, 4b, 6, 7, 10, 11