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LAW OF SINES:

LAW OF SINES:. THE AMBIGUOUS CASE. AMBIGUOUS Open to various interpretations Having double meaning Difficult to classify, distinguish, or comprehend. RECALL: Opposite sides of angles of a triangle Interior Angles of a Triangle Theorem Triangle Inequality Theorem. RECALL:

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LAW OF SINES:

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  1. LAW OF SINES: THE AMBIGUOUS CASE

  2. AMBIGUOUS • Open to various interpretations • Having double meaning • Difficult to classify, distinguish, or comprehend

  3. RECALL: • Opposite sides of angles of a triangle • Interior Angles of a Triangle Theorem • Triangle Inequality Theorem

  4. RECALL: • Oblique Triangles • Triangles that do not have right angles • (acute or obtuse triangles)

  5. RECALL: • LAW OF SINE • – 1  sin  1

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

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

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

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

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

  11. C b a h = b sin A B A c Possible Outcomes Case 2: If A is acute and a > b C b h a  A B 1 SOLUTION

  12. Possible Outcomes Case 3: If A is obtuse and a > b C a b A c B ONE SOLUTION

  13. Possible Outcomes Case 3: If A is obtuse and a ≤ b C a b A c B NO SOLUTION

  14. First, determine what case you have. (ASA, AAS, SSA, etc.) ASA, AAS SSA • Law of Sines • (“plug & chug”) • determine # of solutions • use the Law of Sines • to find them.

  15. For SSA Triangles: • If A < 90° • a < b 1. a < b(sin A) No Solution 2. a = b(sin A) 1 Solution • a > b(sin A) 2 Solution • a ≥ b 1 Solution • If A ≥ 90° • a ≤ b No Solution • a > b 1 Solution

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

  17. sin A = sin B ab 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

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

  19. c < b EXAMPLE 2 Given:ABC where c= 15 inches b = 25 inches mC = 85o 15 < 25 sin 85o c ? b sin C NO SOLUTION CASE (acute) Find mB, mC, and c.

  20. sin A = sin B ab Sin B  1.66032 mB = ? Sin B > 1 NOT POSSIBLE ! Recall:– 1  sin  1 NO SOLUTION CASE

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

  22. sin A = sin B ab Sin B  1.23644 mB = ? Sin B > 1 NOT POSSIBLE ! Recall:– 1  sin  1 NO SOLUTION CASE

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

  24. sin A = sin B ab 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

  25. sin A = sin Cac1 c1 = 51.04 inches sin A = sin Cac2 c = 13.74 inches

  26. EXAMPLE 3 Final Answers: mB1 = 39o mC1 = 116o c1 = 51.04 in. mB2 = 141o mC2 = 14o C2= 13.74 in. TWO – SOLUTION CASE

  27. CLASSWORK: (notebook) 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

  28. Answers:  1)Case 1: mB=49o,mC=96o,c=15.78 Case 2: mB=131o,mC=14o,c=3.84 2)No possible solution. 3)mB=38o,mC=76o,c=15.93

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