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The Law of Sines!

The Law of Sines!. Objective: Be able to use the Law of Sines to find unknowns in triangles!. Homework: Lesson 12.3/1-10, 12-14, 19, 20 Quiz Friday – 12.1 – 12.3. Quick Review:. What does Soh-Cah-Toa stand for?. What kind of triangles do we use this for?. right triangles.

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The Law of Sines!

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  1. The Law of Sines! Objective: Be able to use the Law of Sines to find unknowns in triangles! Homework: Lesson 12.3/1-10, 12-14, 19, 20 Quiz Friday – 12.1 – 12.3

  2. Quick Review: What does Soh-Cah-Toa stand for? What kind of triangles do we use this for? right triangles What if it’s not a right triangle? GASP!! What do we do then??

  3. B c a A C b The Law of Sines: • Note: • capital letters always stand for __________! • lower-case letters always stand for ________! • Use the Law of Sines ONLY when: • you DON’T have a right triangle AND • you know an angle and its opposite side angles sides

  4. The Law of SINES For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles: Can also be written in this form:

  5. Use Law of SINES when ... • AAS - 2 angles and 1 adjacent side • ASA - 2 angles and their included side If 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:

  6. 63° a 29 A C 42 Let’s do some problems!  Ex. 1: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. 79˚ 38˚

  7. T r s 89° 40° 4.8 Ex. 2: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. 51˚

  8. B a. A C Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. c 76˚ 7 37˚ 67˚ b

  9. B A C b. 3.1 96˚ 12 70˚ 14˚ b

  10. Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. 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 .*

  11. B 80° a = 12 c 70° A C b Ex. 3: con’t The angles in a ∆ total 180°, so angle C = 30°. Set up the Law of Sines to find side b:

  12. B 80° a = 12 c 70° 30° A C b = 12.6 Ex. 3: con’t Set up the Law of Sines to find side c:

  13. B 80° a = 12 c = 6.4 70° 30° A C b = 12.6 Ex. 3: Solution Angle C = 30° Side b = 12.6 cm Side c = 6.4 cm Note: We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

  14. Ex. 4: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. B 30° c a = 30 115° C A b 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.

  15. Ex. 4: con’t B 30° c a = 30 115° C A b To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation. We MUST find angle A first because the only side given is side a. The angles in a ∆ total 180°, so angle A = 35°.

  16. Ex. 4: con’t B 30° c a = 30 115° C A 35° b Set up the Law of Sines to find side b:

  17. Ex. 4: con’t B 30° c a = 30 115° C A 35° b = 26.2 Set up the Law of Sines to find side c:

  18. Ex. 4: Solution B 30° c = 47.4 a = 30 115° C A 35° b = 26.2 Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm Note: Use the Law of Sines whenever you are given 2 angles and one side!

  19. AAS • ASA Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions. The Law of Sines

  20. A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is Draw a diagram Draw Then find the Example 3:

  21. Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow. Example 3:

  22. Divide each side by sin Example 3: Law of Sines Cross products Use a calculator. Answer: The length of the shadow is about 75.9 feet.

  23. Your Turn: A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

  24. A 65º x B 15ft 15º C Example: Finding the Height of a Telephone Pole

  25. The Area of a TriangleUsing Trigonometry We can find the area of a triangle if we are given any two sides of a triangle and the measure of the included angle. (SAS)

  26. Area of an Oblique triangle Using two sides and an Angle.

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