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Bell Work

Master the Law of Sines in trigonometry to solve oblique triangles using various cases and examples. Learn how to apply proportions to find angles or sides accurately.

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Bell Work

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  1. x 9” 7” 30° 10” A Bell Work • Find the measure of angle A • Find x.

  2. 9” 7” A Bell work #1 • Solution hypotenuse opposite

  3. x 30° 10” Bell Work #2

  4. Introduction • Given a right triangle, you should feel comfortable using the three basic trig functions( sin, cos, tan) to determine additional information about the triangle.

  5. Law of Sines • Apply Law of Sines to problem-solving situations • Find the area of triangle.

  6. The Law of Sines Law of Sines In any triangle ABC,with sides a, b, and c, Alternative forms are sometimes convenient to use:

  7. Data Required for Solving Oblique Triangles Case 1 One side and two angles known: • SAA or ASA Case 2 Two sides and one angle not included between the sides known: • SSA • This case may lead to more than one solution. Case 3 Two sides and one angle included between the sides known: • SAS Case 4 Three sides are known: • SSS

  8. Law of sines Case 1 One side and two angles known: • SAA or ASA Case 2 Two sides and one angle not included between the sides known: • SSA • This case may lead to more than one solution.

  9. Example 1

  10. 63° x 47° 5.45” Law of SinesExample 1 • Given the diagram below, determine the length of side x.

  11. 63° x 47° 5.45” Law of SinesExample 1 • What do we know about this problem? • First of all, it is an oblique triangle. • Second, we note that two angles are known, and one of the sides opposite.

  12. 63° x 47° 5.45” Law of SinesExample 1 • That’s enough info to verify that using the Law of Sines will allow us to determine the length of x.

  13. 63° x 47° 5.45” Law of SinesExample 1 • To solve, set up a proportion. Remember that the sides are proportional to the sines of the opposite angles.

  14. 63° x 47° 5.45” Law of SinesExample 1 • Start by pairing the 63° angle and the 5.45” side together since they are opposite one another.

  15. 63° x 47° 5.45” Law of SinesExample 1 • The unknown side x is opposite the 47° angle. Pair these up to complete the proportion.

  16. 63° x 47° 5.45” Law of SinesExample 1 • Solve the proportion by cross-multiplying. Multiply on this diagonal first. 5.45 x sin47° = 3.99

  17. 63° x 47° 5.45” Law of SinesExample 1 • Solve the proportion by cross-multiplying. Next, divide 3.99 by sin63° 3.99 ÷ sin63° = 4.47 (this is the length of x)

  18. 63° 47° 5.45” Law of SinesExample 1 • By using the Law of Sines, we know the length of side x is 4.47 inches. 4.47”

  19. Example 2

  20. 85° 42° x 65.85 mm Law of SinesExample 2 • Given the diagram below, determine the length of side x.

  21. 85° 42° x 65.85 mm Law of SinesExample 2 • Before you jump in, be sure you know what you are dealing with. • You are working with an oblique triangle... • …and you know two angles and a side opposite one of those angles.

  22. 85° 42° x 65.85 mm Law of SinesExample 2 • That means using the Law of Sines will allow you to solve for x.

  23. 85° 42° x 65.85 mm Law of SinesExample 2 • Set-up a proportion, starting with the 65.85 mm side and the 85° angle since they are opposite one another.

  24. 85° 42° x 65.85 mm Law of SinesExample 2 • Solve the proportion. Multiply on this diagonal first. 65.85 x sin42° = 44.1

  25. 85° 42° x 65.85 mm Law of SinesExample 2 • Solve the proportion. Multiply on this diagonal first. 65.85 x sin42° = 44.1

  26. 85° 42° x 65.85 mm Law of SinesExample 2 • Solve the proportion. Next, divide 44.1 by sin85° 44.1 ÷ sin85° = 44.2 (this is the length of x)

  27. Law of SinesExample 2 • Using the Law of Sines on this problem gives you an answer of 44.2 mm. 85° 42° 44.2 mm 65.85 mm

  28. Example 3

  29. Law of SinesExample 3 • Try this one on your own. • Set-up a proportion and solve for x. 9.25 cm 88° x 57°

  30. Law of SinesExample 3 • How did it turn out? x = 11.02 cm 9.25 cm 88° x 57°

  31. Example 4

  32. 85.5 mm 70 mm 42° Law of Sines • Recall that the other scenario where you can use the Law of Sines is when you know the lengths of two sides and the size of an angle opposite on of those sides.

  33. 85.5 mm 70 mm 42° Law of SinesExample 4 • Given the diagram below, determine the size of angle A. A

  34. 85.5 mm 70 mm 42° Law of SinesExample 4 • Once again, set-up a proportion. • Start by pairing-up the 70 mm side and the 42° angle. A

  35. 85.5 mm 70 mm 42° Law of SinesExample 4 • Complete the proportion by putting the 85.5 mm side and angle A together. A

  36. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Cross-multiply on this diagonal...

  37. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Cross-multiply on this diagonal...

  38. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Cross-multiply on this diagonal...

  39. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Divide both sides by 70.

  40. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Evaluate the left side of the equation.

  41. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Evaluate the left side of the equation.

  42. Law of SinesExample 4 • This proportion will be a little more difficult to solve. The steps are shown below: Type 0.8173 into your calculator, press the 2nd function key, then press the sin key.

  43. 85.5 mm 70 mm 42° Law of SinesExample 4 • So the size of angle A is 54.8°. 54.8°

  44. Example 5

  45. A 4.2” 4.9” 55° Law of SinesExample 5 • Given the diagram below, determine the size of angle A.

  46. A 4.2” 4.9” 55° Law of SinesExample 5 • Once again, set-up a proportion. • Start by pairing-up the 4.2” side and the 55° angle.

  47. A 4.2” 4.9” 55° Law of SinesExample 5 • Complete the proportion by putting the 4.9” side and angle A together.

  48. Law of SinesExample 5 • Follow the steps shown to solve the proportion: Cross-multiply on this diagonal...

  49. Law of SinesExample 5 • Follow the steps shown to solve the proportion: …then multiply on this diagonal...

  50. Law of SinesExample 5 • Follow the steps shown to solve the proportion: Divide both sides by 4.2.

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