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4-7 Ambiguous Case for Law of Sines and Law of Cosines

4-7 Ambiguous Case for Law of Sines and Law of Cosines. Chapter 4 Trigonometric Functions. Warm-up: Use the Law of Sines and area formula. Solve. A = 41 , B = 49, a = 6.5 Find the area. a = 8.4, b = 10, C = 108 . Objectives.

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4-7 Ambiguous Case for Law of Sines and Law of Cosines

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  1. 4-7 Ambiguous Case for Law of SinesandLaw of Cosines Chapter 4 Trigonometric Functions

  2. Warm-up: Use the Law of Sines and area formula • Solve. A = 41, B = 49, a = 6.5 • Find the area. a = 8.4, b = 10, C = 108

  3. Objectives • Determine whether a triangle has zero, one, or two solutions. • Solve ambiguous triangles using Law of Sines. • Solve triangles using Law of Cosines.

  4. Recall that in Geometry… …the 5 ways to prove that two triangles are congruent. SAS ASA AAS SSS HL Special case for SSA Usually you can’t use SSA. Here’s why!

  5. Case 1 When the angle A is acute • If side a is smaller than side b, three possibilities exist. 1. Two solutions C b aa  BB A

  6. Case 2 When the angle A is acute • If side a is smaller than side b, three possibilities exist. 2. One solution (This is why the HL can prove congruence). C b a  B A

  7. Case 3 When the angle A is acute • If side a is smaller than side b, three possibilities exist. 3. No solutions, if one side is too short to make a triangle. C a b  B A

  8. Case 4 When the angle A is acute • If side a is greater than or equal to side b, one possibility exists. C b a  A B

  9. Case 5 When A  90 if a  b, then a is too short to reach side c. C a b  A

  10. Case 6 When A  90 if a > b, then a solution is possible. C b a  A B

  11. Hint: Always check for two solutions when given the measures of two sides and a non-included angle that is acute. (SSA) Example 1 • Determine the number of possible solutions for each triangle. • A = 63, a = 18, b = 25 Angle A is acute, and 18 < 25, consider Cases 1 – 4.

  12. Hint: Always check for two solutions when given the measures of two sides and a non-included angle that is acute. (SSA) Example 2 • Solve for two triangles for which A = 45°, a = 18, and c = 24.

  13. Example 3 • Determine the number of possible solutions for each triangle. b) A = 105, a = 73, b = 55 Since 105  90, consider Cases 5 and 6.

  14. Example 4 • Find all solutions for each triangle. If no solutions exist, write none. a) A = 98, a = 39, b = 22

  15. Example 5 • Find all solutions for each triangle. If no solutions exist, write none. b) A = 72.2, a = 21, b = 22

  16. Deriving the Law of Cosines B c a h A x D b - x C b

  17. Example 5: Solve B 14 a A 12 C

  18. Assignment: • P. 298, 2, 3, 6, 10 – 12, 19.

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