The Ambiguous Case for the Law of Sines 5.7 The Ambiguous Case for the Law of Sines

AMBIGUOUS Open to various interpretations Having double meaning Difficult to classify, distinguish, or comprehend /ctr

Opposite sides of angles of a triangle RECALL: Opposite sides of angles of a triangle Interior Angles of a Triangle Theorem Triangle Inequality Theorem /ctr

Triangles that do not have right angles RECALL: Oblique Triangles Triangles that do not have right angles (acute or obtuse triangles) /ctr

RECALL: LAW OF SINE – 1  sin   1 /ctr

Sine values of supplementary angles are equal. RECALL: Sine values of supplementary angles are equal. Example: Sin 80o = 0.9848 Sin 100o = 0.9848 /ctr

Law of Sines: The Ambiguous Case Given: lengths of two sides and the angle opposite one of them (S-S-A) /ctr

Possible Outcomes Case 1: If A is acute and a < b a. If a < b sinA a C b a h = b sin A b B A h c A B c NO SOLUTION

Possible Outcomes Case 1: If A is acute and a < b b. If a = b sinA h = b sin A b h = a A c B 1 SOLUTION

Possible Outcomes Case 1: If A is acute and a < b h = b sin A b. If a > b sinA C b a h a   180 -  A B B c 2 SOLUTIONS

Possible Outcomes Case 2: If A is obtuse and a > b C a b A c B ONE SOLUTION

Possible Outcomes Case 2: If A is obtuse and a ≤ b C a b A c B NO SOLUTION

Determine the number of possible solutions for each triangle. i) A=30deg a=8 b=10 ii) b=8 c = 10 B = 118 deg

Find all solutions for each triangle. i) a = 4 b = 3 A = 112 degrees ii) A = 51 degrees a = 40 c = 50

EXAMPLE 1 Given: ABC where a = 22 inches b = 12 inches a>b mA = 42o a>b mA > mB SINGLE–SOLUTION CASE (acute) Find m B, m C, and c.

sin A = sin B a b Sin B  0.36498 mB = 21.41o or 21o Sine values of supplementary angles are equal. The supplement of B is B2.  mB2=159o

mC = 180o – (42o + 21o) mC = 117o sin A = sin C a c c = 29.29 inches SINGLE–SOLUTION CASE

sin A = sin B a b Sin B  1.66032 mB = ? Sin B > 1 NOT POSSIBLE ! Recall: – 1  sin   1 NO SOLUTION CASE /ctr

EXAMPLE 3 Given: ABC where b = 15.2 inches a = 20 inches b < a mB = 110o b < a NO SOLUTION CASE (obtuse) Find m B, m C, and c.

sin A = sin B a b Sin B  1.23644 mB = ? Sin B > 1 NOT POSSIBLE ! Recall: – 1  sin   1 NO SOLUTION CASE /ctr

EXAMPLE 4 Given: ABC where a = 24 inches b = 36 inches a < b mA = 25o a < b a ? b sin A 24 > 36 sin 25o TWO – SOLUTION CASE (acute) Find m B, m C, and c.

sin A = sin B a b Sin B  0.63393 mB = 39.34o or 39o The supplement of B is B2.  mB2 = 141o mC1 = 180o – (25o + 39o) mC1 = 116o mC2 = 180o – (25o+141o) mC2 = 14o

sin A = sin C a c1 c1 = 51.04 inches sin A = sin C a c2 /ctr

EXAMPLE 3 Final Answers: mB1 = 39o mC1 = 116o c1 = 51.04 in. TWO – SOLUTION CASE /ctr

Find m B, m C, and c, if they exist. SEATWORK: (notebook) Answer in pairs. Find m B, m C, and c, if they exist.  1) a = 9.1, b = 12, mA = 35o  2) a = 25, b = 46, mA = 37o 3) a = 15, b = 10, mA = 66o   /ctr

Answers: 1)Case 1: mB=49o,mC=96o,c=15.78 Case 2: 2)No possible solution. 3)mB=38o,mC=76o,c=15.93   /ctr