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6-1 law of sines

Chapter 6. 6-1 law of sines. Use the law of sines to solve triangles. Objectives. For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:. Law of sines.

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6-1 law of sines

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  1. Chapter 6 6-1 law of sines

  2. Use the law of sines to solve triangles Objectives

  3. For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles: Law of sines

  4. you have 3 dimensions of a triangle and you need to find the other 3 dimensions - they cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation. Use the Law of Sines if you are given: AAS - 2 angles and 1 adjacent sideor ASA- 2 angles and their included side SSS- three sides SSA(this is an ambiguous case)

  5. You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c. * In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .* Example #1

  6. B 80° a = 12 c 70° A C b The angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b: solution

  7. B:

  8. You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c. Example#2

  9. B 30° c a = 30 115° C A b solution

  10. solution

  11. For the triangle in fig 6.3 C=102.3 degree, B=28.7 degree and b=27.4 feet C A B Check it out

  12. When given SSA (two sides and an angle that is NOT the included angle) , the situation is ambiguous. The dimensions may not form a triangle, or there may be 1 or 2 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist. The Ambiguous Case (SSA)

  13. C = ? ‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position b A B ? c = ? In the following examples, the given angle will always be angle A and the given sides will be sides a and b. If you are given a different set of variables, feel free to change them to simulate the steps provided here. Example

  14. C = ? a b A B ? c = ? Situation I: Angle A is obtuse If angle A is obtuse there are TWO possibilities If a ≤ b, then a is too short to reach side c - a triangle with these dimensions is impossible. Ambiguous case

  15. C = ? a b A B ? c = ? If a > b, then there is ONE triangle with these dimensions.

  16. C a = 22 15 = b 120° A B c Situation I: Angle A is obtuse - EXAMPLE Given a triangle with angle A = 120°, side a = 22 cm and side b = 15 cm, find the other dimensions. Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines:

  17. solution Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3 cm

  18. C = ? b a A B ? c = ? Situation II: Angle A is acute If angle A is acute there are SEVERAL possibilities. Side ‘a’ may or may not be long enough to reach side ‘c’. We calculate the height of the altitude from angle C to side c to compare it with side a.

  19. h C = ? b a A B ? c = ? Situation II: Angle A is acute First, use SOH-CAH-TOA to findh: Then, compare ‘h’ to sides a and b . . .

  20. C = ? a b h A B ? c = ? If a < h, then NO triangle exists with these dimensions. Different Triangles

  21. C C b b a h a h A B A c c B If h < a < b, then TWO triangles exist with these dimensions. If we open side ‘a’ to the outside of h, angle B is acute If we open side ‘a’ to the inside of h, angle B is obtuse.

  22. C b a h A B c If h < b < a, then ONE triangle exists with these dimensions. Since side a is greater than side b, side a cannot open to the inside of h, it can only open to the outside, so there is only 1 triangle possible!

  23. C b a = h A c B If h = a, then ONE triangle exists with these dimensions. If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions.

  24. C = ? a = 12 15 = b h 40° A B ? c = ? Given a triangle with angle A = 40°, side a = 12 cm and sideb = 15 cm, find the other dimensions. Example

  25. Angle B = 53.5° Angle C = 86.5° Side c = 18.6 Angle B = 126.5° Angle C = 13.5° Side c = 4.4 solution

  26. if a < b no solution if angle A is obtuse if a > b one solution (Ex I) if a < h no solution (Ex II-1) if h < a < b 2 solutions one with angle B acute, one with angle B obtuse if angle A is acute find the height, h = b*sinA (Ex II-2) if a > b > h 1 solution If a = h 1 solution angle B is right The Ambiguous Case - Summary

  27. Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions. AAS ASA SSA (the ambiguous case) Law of sines

  28. Do problems 7-12 in your book page 410 Student guided practice

  29. Do problems 13-17, 27-29 in your book page 410 Homework

  30. Today we learned about the law of sines Next class we are going to learn about law of cosines closure

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