8_TheLawofCosines.pptx 8_TheLawofCosines.pptx

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1. The Law of Cosines

2. Warm Up Use the figure to find each ratio. 1. sin J 2. cos L 3. tan L 4. sin L 5. cos J 6. tan J

3. Warm Up Use the figure to find each ratio. 1. sin J 2. cos L 3. tan L 4. sin L 5. cos J 6. tan J  

4. Standards for Mathematical Content G.SRT.10  Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11  Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

5. MP3 Construct viable arguments and critique the reasoning of others. MP6 Attend to precision. Standards for Mathematical Practice

6. Lesson Goal ● Understand and apply the Law of Cosines to find unknown measurements in right and nonright triangles.

7. Learn The Law of Cosines When the Law of Sines cannot be used to solve a triangle, the Law of Cosines may apply. You can use the Law of Cosines to find the length of the third side of a triangle when the measures of two sides and the included angle are known or to find the angle measures of a triangle if the lengths of all three sides are known.

8. Learn The Law of Cosines You can use the Law of Cosines to solve a triangle if you know the measures of two sides and the included angle (SAS). Theorem 9.11: Law of Cosines If has lengths , , and , representing the lengths of the sides opposite the angles with measures , , and , then Theorem 9.11: Law of Cosines

9. Learn The Law of Cosines Talk About It! What happens when you apply the Law of Cosines to find the missing measures of a right triangle?

10. Example 1 The Law of Cosines (SAS) Find the value of to the nearest tenth.  

11. Example 1 The Law of Cosines (SAS) Think About It! Why can’t you use the Law of Sines to find the value of ?  

12. Example 1 The Law of Cosines (SAS) Law of Cosines Substitution Simplify. Take the positive square root of each side. Use a calculator. Law of Cosines Substitution Simplify. Take the positive square root of each side. Use a calculator. We are given the measures of two sides and their included angle. Use the Law of Cosines to write an equation.

13. Example 2 The Law of Cosines (SSS) You can also use the Law of Cosines if you know the three side lengths.

14. Example 2 The Law of Cosines (SSS) Find the value of to the nearest whole number.  

15. Example 2 The Law of Cosines (SSS) Law of Cosines Substitution Simplify. Subtract from each side. Divide each side by . Use the inverse cosine function. Use a calculator. Law of Cosines Substitution Simplify. Use the inverse cosine function. Use a calculator.

16. Example 2 The Law of Cosines (SSS) Check Find the value of to the nearest whole number.  

17. Example 2 The Law of Cosines (SSS) Check Find the value of to the nearest whole number.  

18. Example 3 Indirect Measurement with the Law of Cosines GOLF Nhat is golfing and uses a distance measuring tool to determine that the tee box where he is standing is yards from the hole. To avoid a water hazard, Nhat turns and hits a shot yards up the fairway. Complete the diagram with the correct values. Then find the distance between Nhat’s ball and the hole to the nearest yard.  

19. Example 3 Indirect Measurement with the Law of Cosines Because we know the measures of two sides of the triangle and the included angle, use the Law of Cosines to find the remaining distance.

20. Example 3 Indirect Measurement with the Law of Cosines Law of Cosines Simplify. Take the positive square root of each side. Use a calculator. Law of Cosines Simplify. Take the positive square root of each side. Use a calculator. Nhat’s ball is about yards from the hole.  

21. Example 3 Indirect Measurement with the Law of Cosines Check CELL PHONE TOWERS A cell phone company builds two towers that are  miles apart. They choose a random location miles from Tower A and  miles from Tower B to test the towers’ signal strengths. Find the value of to the nearest degree.  

22. Example 3 Indirect Measurement with the Law of Cosines Check CELL PHONE TOWERS A cell phone company builds two towers that are  miles apart. They choose a random location miles from Tower A and  miles from Tower B to test the towers’ signal strengths. Find the value of to the nearest degree.  

23. Example 4 Solve a Nonright Triangle with the Law of Cosines When solving right triangles, you can use sine, cosine, or tangent. When solving any triangle, you can use the Law of Sines or the Law of Cosines, depending on what information is given.

24. Example 4 Solve a Nonright Triangle with the Law of Cosines Solve . Round to the nearest degree.  

25. Example 4 Solve a Nonright Triangle with the Law of Cosines Because this is not a right triangle. The measures of all three sides are given (SSS), so decide which angle measure you want to find. Then use the Law of Cosines.  

26. Example 4 Solve a Nonright Triangle with the Law of Cosines Law of Cosines Substitute. Simplify. Subtract from each side. Divide each side by . Use the inverse cosine function. Use a calculator. Law of Cosines Substitute. Simplify. Use the inverse cosine function. Use a calculator.

27. Example 4 Solve a Nonright Triangle with the Law of Cosines Use the Law of Sines to find .   Law of Sines and Multiplication Property of Equality Divide each side by . Use the inverse sine function. Use a calculator. Law of Sines Multiplication Property of Equality Use the inverse sine function. Use a calculator.

28. Example 4 Solve a Nonright Triangle with the Law of Cosines By the Triangle Angle-Sum Theorem, or .  

29. Example 4 Solve a Nonright Triangle with the Law of Cosines Check Solve when , , and Round angle measures to the nearest degree and side measures to the nearest tenth.  

30. Example 4 Solve a Nonright Triangle with the Law of Cosines Check Solve when , , and Round angle measures to the nearest degree and side measures to the nearest tenth.  

31. Example 5 Solve a Right Triangle with the Law of Cosines Solve . Round angle measures to the nearest degree and side measures to the nearest tenth.  

32. Example 5 Solve a Right Triangle with the Law of Cosines Find .   Pythagorean Theorem Substitution Simplify. Take the positive square root of each side. Use a calculator. Pythagorean Theorem Substitution Simplify. Take the positive square root of each side. Use a calculator. So, .  

33. Example 5 Solve a Right Triangle with the Law of Cosines Find .   Law of Cosines Substitution Simplify. Subtract from each side. Divide each side by . Use a calculator. Law of Cosines Substitution Simplify. Use a calculator. So, .  

34. Example 5 Solve a Right Triangle with the Law of Cosines Find . Because we know that and , by the Triangle Angle-Sum Theorem, or .  

35. Example 5 Solve a Right Triangle with the Law of Cosines Check Solve . Round angle measures to the nearest degree and side measures to the nearest tenth.  

36. Example 5 Solve a Right Triangle with the Law of Cosines Check Solve . Round angle measures to the nearest degree and side measures to the nearest tenth.  

37. Exit Ticket Can you solve any triangle from the given information? Answer yes or no . If yes , describe your method. 1. two angle measures and a side length (AAS or ASA) 2. two side lengths and an angle measure (SAS) 3. two side lengths and an angle measure (SSA)

38. Exit Ticket Can you solve any triangle from the given information? Answer yes or no . If yes , describe your method. 4. three angle measures 5. three side lengths

39. Exit Ticket Can you solve any triangle from the given information? Answer yes or no . If yes , describe your method. 1. two angle measures and a side length (AAS or ASA) yes; Law of Sines 2. two side lengths and an angle measure (SAS) yes; Law of Cosines 3. two side lengths and an angle measure (SSA) yes; Law of Sines (might be ambiguous)

40. Exit Ticket Can you solve any triangle from the given information? Answer yes or no . If yes , describe your method. 4. three angle measures no 5. three side lengths yes; Law of Cosines (Law of Sines can be used after largest angle is found).