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Warm up

Warm up.   A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest degree)? Cos -1 (1.9/5.2) = 69 o. The Law of Sines.

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Warm up

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  1. Warm up   A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to the nearest degree)? Cos-1(1.9/5.2) = 69o

  2. The Law of Sines

    Objective: To use the Law of Sines in order to solve oblique triangles
  3. Consider the first category, an acute triangle (, ,  are acute).
  4. Create an altitude, h.
  5. Theorem Law of Sines
  6. The Law of Sines is used when we know any two angles and one side or when we know two sides and an angle opposite one of those sides.
  7. Applying the Law of Sines The Law of Sines may be used when the known parts of the triangle are: 1. one side and two angles (SAA) 2. two sides and an angle opposite one of the sides (SSA)
  8. Example: In triangle ABC, angle A = 106 o, angle B = 31o and side a = 10 cm. Solve the triangle ABC by finding angle C and sides b and c.(round answers to 1 decimal place).
  9. Solution Use the fact that the sum of all three angles of a triangle is equal to 180 o to write an equation in C. A + B + C = 180 o Solve for C. C = 180 o - (A + B) = 43 o
  10. Solution (cont’d) Use sine law to write an equation in b. a / sin(A) = b / sin(B) Solve for b. b = a sin (B) / sin(A) = (approximately) 5.4 cm Use the sine law to write an equation in c.a / sin(A) = c / sin(C)Solve for c. c = a sin (C) / sin(A) = (approximately) 7.1 cm
  11. Warm up Using the triangle above, A = 50o, B = 65o and a = 12.  Solve the triangle.
  12. Law of Sines practice http://www.emathematics.net/trigonometria.php?tr=5
  13. The Ambiguous Case – SSA In this case, you may have information that results in one triangle, two triangles, or no triangles.
  14. SSA – No Solution Two sides and an angle opposite one of the sides.
  15. By the law of sines,
  16. Thus, Therefore, there is no value for  that exists! No Solution!
  17. SSA – Two Solutions
  18. By the law of sines,
  19. So that,
  20. Case 1Case 2 Both triangles are valid! Therefore, we have two solutions.
  21. Case 1
  22. Case 2
  23. Practice Solve the triangle: A = 42°, a = 11, and b = 6
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