Understanding the Ambiguous Case of Sine/Cosine Law
Explore the ambiguous case of the sine law and cosine law, where more than one triangle solution is possible. Learn how to analyze and solve problems involving this scenario. The concept of ambiguity in geometry is illustrated through practical examples and scenarios.
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Presentation Transcript
4.3 The Ambiguous Case of the Sine/Cosine Law Oct 27th2017
Lesson Objective Analyze the ambiguous case of the sine law, and solve problems that involve the ambiguous case
Ambiguous The word Ambiguous means something that can have more than one interpretation, More than one answer.
Consider the Following Martina and Carl are both holding a rope that attached to a balloon in the sky. Martina s rope is 7.8 m long and makes a 36oangle with the ground Carl s rope is 5.8 m long How far are Martina and Carl away from each other?
Consider the Following On your sheet, there is an unfinished triangle. With a ruler, draw a 5.8 cm line from point A to the ground. Then using a protractor, measure the angle. Is there only one solution? In other words, is there only one triangle that can be made?
7.8 M 5.8 m 5.8 m 36O
7.8 M 5.8 m 5.8 m 36O 50o 130 o
7.8 M 5.8 m 5.8 m 130o 50o 36o Triangle 1 Triangle 2 Sin 130o 7.8 m Sin 50o 7.8 m Sin 36o 5.8 m Sin 36o 5.8 m = =
BOTH TRIANGLES ARE POSSIBLE! 7.8 M 5.8 m 5.8 m 130o 50o 36o Triangle 1 Triangle 2 0.766 7.8 m 0.766 7.8 m Sin 36o 5.8 m Sin 36o 5.8 m = =
Summary In problems when you are given two sides and an angle (SSA), there can be more than one triangle formed.
There are Four SSA Scenarios SCENARIO 1 The side is not long enough No triangle is possible 12 m 4 m 30o
There are Four SSA Scenarios SCENARIO 2 The side length is JUST right One triangle is possible 12 m 6 m 30o
There are Four SSA Scenarios SCENARIO 3 The side length is longer than 6 m, but shorter than 12 Two triangles are possible 12 m 8 m 8 m 30o
There are Four SSA Scenarios SCENARIO 4 The side length is longer than 12 m Only ONE triangle is possible 12 m 15 m 30o
Example Problem #1 Given the following measurements for triangle ABC, determine if there are zero, one or two possibilities. A = 75o a = 4 m b = 12 m
Example Problem #2 Decide whether each description of a triangle involves the SSA situation In triangle DEF, D = 81o, e = 9 cm, and f = 8 cm
Homework Page 183 Questions #1 a-d , 2 a- f and 3
