4 3 The Ambiguous Case of the SineCosine

4. 3 The Ambiguous Case of the Sine/Cosine Law Oct 27 th 2017

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 36 o angle 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 36 O 5. 8 m

7. 8 M 36 O 5. 8 m 130 o 5. 8 m 50 o

7. 8 M 36 o Triangle 1 Sin 130 o Sin 36 o = 7. 8 m 5. 8 m 130 o 5. 8 m 50 o Triangle 2 Sin 50 o Sin 36 o = 7. 8 m 5. 8 m

BOTH TRIANGLES ARE POSSIBLE! 7. 8 M 36 o Triangle 1 0. 766 Sin 36 o = 7. 8 m 5. 8 m 130 o 5. 8 m 50 o Triangle 2 0. 766 Sin 36 o = 7. 8 m 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 30 o 4 m

There are Four SSA Scenarios SCENARIO 2 The side length is JUST “right” One triangle is possible 12 m 30 o 6 m

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 30 o 8 m 8 m

There are Four SSA Scenarios SCENARIO 4 The side length is longer than 12 m Only ONE triangle is possible 12 m 30 o 15 m

Example Problem #1 Given the following measurements for triangle ABC, determine if there are zero, one or two possibilities. ∠ A = 75 o a=4 m m b = 12

Example Problem #2 Decide whether each description of a triangle involves the SSA situation In triangle DEF, ∠ D = 81 o, e = 9 cm, and f = 8 cm

Homework Page 183 Questions #1 a-d , 2 a- f and 3