7_TheLawofSines.pptx TheLawofSines.pptx g

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1. The Law of Sines

2. Warm Up Solve each proportion. 1. 2. 3. 4. 5. MIXTURES Braden uses teaspoons of fertilizer for each gallon of water. How many gallons of the mixture can he make with teaspoons of fertilizer?  

3. Warm Up Solve each proportion. 1. 2. 3. 4. 5. MIXTURES Braden uses teaspoons of fertilizer for each gallon of water. How many gallons of the mixture can he make with teaspoons of fertilizer? gal  

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. MP4 Model with mathematics. MP5 Use appropriate tools strategically. Standards for Mathematical Practice

6. Lesson Goals ● Understand and apply the Law of Sines to find unknown measurements in right and nonright triangles. ● Determine whether three given measures of a triangle define 0, 1, or 2 triangles by using the Law of Sines.

7. Learn The Law of Sines The Law of Sines can be used to find side lengths and angle measurements for any triangle. Theorem 9.10: Law of Sines If has lengths , , and , representing the lengths of the sides opposite the angles with measures , , and , then Theorem 9.10: Law of Sines You can also use the Law of Sines to solve a triangle if you know the measures of two angles and any side (AAS or ASA).

8. Example 1 The Law of Sines (AAS) Find the value of to the nearest tenth.  

9. Example 1 The Law of Sines (AAS) Because we are given the measures of two angles and a nonincluded side, use the Law of Sines to write a proportion. Law of Sines Cross Products Property Divide each side by . Use a calculator. Law of Sines Cross Products Property Use a calculator.

10. Example 1 The Law of Sines (AAS) Think About It! How could you find the value of ?  

11. Example 1 The Law of Sines (AAS) Check Find the value of to the nearest tenth.    

12. Example 1 The Law of Sines (AAS) Check Find the value of to the nearest tenth.    

13. Example 2 The Law of Sines (ASA) Find the value of to the nearest tenth.  

14. Example 2 The Law of Sines (ASA) By the Triangle Angle-Sum Theorem, Law of Sines Cross Products Property Divide each side by sin . Use a calculator. Law of Sines Cross Products Property Use a calculator.

15. Example 2 The Law of Sines (ASA) Think About It! Could you use to find the value of ? Justify your reasoning.  

16. Example 2 The Law of Sines (ASA) Check Find the value of to the nearest tenth. A. B. C. D.  

17. Example 2 The Law of Sines (ASA) Check Find the value of to the nearest tenth. A. B. C. D.  

18. Example 3 Indirect Measurement with the Law of Sines SURVEYING Mr. Fortunado is having a boundary survey done on his property. What is the distance between Mr. Fortunado’s home and his neighbor’s?

19. Example 3 Indirect Measurement with the Law of Sines Because we know two angles of a triangle and one nonincluded side, use the Law of Sines. Law of Sines Cross Products Property Use a calculator. Law of Sines Cross Products Property Use a calculator.

20. Learn The Ambiguous Case If you are given the measures of two angles and a side, exactly one triangle is possible. However, if you are given the measures of two sides and the angle opposite one of them, zero, one, or two triangles may be possible. This is known as the ambiguous case . So, when solving a triangle using the SSA case, zero, one, or two solutions are possible.

21. Learn The Ambiguous Case Key Concept: Possible Triangles in SSA Case Consider a triangle in which , , and are given and is the altitude of the triangle. Below are the triangles that are possible when is acute . Angle A is acute. no solution one solution two solutions one solution Key Concept: Possible Triangles in SSA Case Angle A is acute.

22. Learn The Ambiguous Case Angle A is right or obtuse. no solution no solution one solution one solution Angle A is right or obtuse.

23. Learn The Ambiguous Case Solving a triangle with an obtuse angle sometimes requires finding sine ratios for measures greater than . The sine ratios for obtuse angles are defined based on their supplementary angles.   Postulate 9.1 Words The sine of an obtuse angle is defined to be the sine of its supplement. Symbols If and then Postulate 9.1 Words The sine of an obtuse angle is defined to be the sine of its supplement. Symbols

24. Example 4 The Ambiguous Case with One Solution Because , you can use to find in acute triangles. In , and . Determine whether has no solution, one solution, or two solutions. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.  

25. Example 4 The Ambiguous Case with One Solution Because is acute, and , you know that one solution exists. Step 1 Use the Law of Sines to find .   Law of Sines Multiply each side by . Law of Sines Use the function to find the exact value of or about  

26. Example 4 The Ambiguous Case with One Solution Step 2 Use the Triangle Angle-Sum Theorem to find .    

27. Example 4 The Ambiguous Case with One Solution Step 3 Use the Law of Sines to find .   Law of Sines Solve for . Use a calculator. Law of Sines Use a calculator. So,  

28. Example 4 The Ambiguous Case with One Solution Talk About It! If the given angle is a right angle and there is one solution to the triangle, how can you find the third side?

29. Example 5 The Ambiguous Case with No Solution In , , , and . Determine whether has no solution, one solution, or two solutions. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.  

30. Example 5 The Ambiguous Case with No Solution Because is obtuse, and , there is no solution.  

31. Example 5 The Ambiguous Case with No Solution Think About It! If were acute and , how could you find the number of possible solutions?  

32. Example 6 The Ambiguous Case with More than One Solution In , , , and . Determine whether has no solution, one solution, or two solutions. Then, solve the triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree.  

33. Example 6 The Ambiguous Case with More than One Solution Because is acute, and , find and compare it to .   Use a calculator. Use a calculator. Because or there are two solutions.  

34. Example 6 The Ambiguous Case with More than One Solution is acute. is obtuse. Find Find Law of Sines Find an obtuse angle B for which sin   B   0.6359. Solve for Postulate 9.1 or Use a calculator. Use the function. Law of Sines Use a calculator.

35. Example 6 The Ambiguous Case with More than One Solution Find Find or or Find c . Find c . Law of Sines Law of Sines Solve for . Solve for c . Use a calculator. Use a calculator. Find c . Find c . Law of Sines Law of Sines Solve for c . Use a calculator. Use a calculator.

36. Example 6 The Ambiguous Case with More than One Solution So, one solution is , , and c 26.8, and another solution is , , and c 3.4.  

37. Exit Ticket If you are given one of the following and want to solve a triangle, which one is ambiguous? a. two angle measures and any side length b. three side lengths c. two side lengths and the measure of the angle opposite one side d. two side lengths and the measure of the included angle

38. Exit Ticket If you are given one of the following and want to solve a triangle, which one is ambiguous? C a. two angle measures and any side length b. three side lengths c. two side lengths and the measure of the angle opposite one side d. two side lengths and the measure of the included angle