Chapter 5: Trigonometric Functions Lesson: Ambiguous Case in Solving Triangles Mrs. Parziale

Do Now How long is the side opposite the 32° angle of a right triangle if the hypotenuse is 10 cm?

Categories of Triangles SSS – 3 sides known – Law of Cosines to find largest angle – Law of Sines to find second angle – Property of Angles of a Triangle to find third angle SAS – 2 sides and 1 included angle known – Law of Cosines to find the third side – Law of Sines to find second angle – Property of Angles of a Triangle to find third angle

Categories of Triangles ASA – two angles and an included side are known – Property of Angles of a Triangle to find third angle – Law of Sines (twice) to find remaining two sides AAS – two angles and a non-included side are known. – known side must be opposite one of the known angles – Property of Angles of a Triangle to find third angle – Law of Sines (twice) to find remaining two sides

SSA – Ambiguous Case Two sides and a non-included angle are known Six possible configurations result. – Two cases if the given angle is obtuse or right – Four cases if the given angle is acute

Given Angle  B is obtuse or right: 1. If, then one solution is possible Example 1: Given the following triangle – B a b

Given Angle  B is obtuse or right: 2. If, then no solutions are possible. Example 2: Given the following triangle – B a b

Given Angle  B is acute: In the figure below, h is the altitude, where. 1. If, then no solution is possible. – That is,. Example 3: Given the following triangle – B a b h

Given Angle  B is acute: In the figure below, h is the altitude, where. 2. If, then exactly one right triangle is formed. Example 4: Given the following triangle – B b = h a

Given Angle  B is acute: 4.If, then only one solution exists. Example 6: Given the following triangle – If then the solution is an isosceles triangle. Example 7: Given the following triangle – B a b h

Given Angle  B is acute: 3. If, then two different solutions are possible. (That is,.) Example 5: Given the following triangle – B a b h

Closure Solve the triangle with the given measurements: