1. LESSON 10-1: THE PYTHAGOREAN THEOREM

2. SIMPLIFYING RADICALS LESSON 10-2

3. ESSENTIAL UNDERSTANDING A radical is any number with a square root. You can simplify a radical expressions using multiplication and division. MULIPLICATION PROPERTY OF SQUARE ROOTS: √32 = √16 ∙ √2 = 4√2

4. PROBLEM 1 What is the simplified form of √160? Ask “what perfect square goes into 160?” 160 = 2 ∙ 80 No perfect square 160 = 4 ∙ 40Yes perfect square 160 = 16 ∙ 10Yes perfect square √160 = √16 ∙ √10 = 4√10

5. GOT IT? 1 Simplify: √72

6. PROBLEM 2 Simplify: √54n 7 √9 ∙ 6 ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n ∙ n √9 ∙ √6 ∙ √n 2 ∙ √n 2 ∙ √n 2 ∙ √n 3 ∙ √6 ∙ n ∙ n ∙ n ∙ √n 3n 3 ∙ √6n 3n 3 √6n

7. GOT IT? 2 Simplify: -m √80m 9

8. PROBLEM 3 Simplify: 2√7t ∙ 3√14t 2 2 ∙ 3 ∙ √ 7t ∙ 14t 2 6 ∙ √98t 3 6 ∙ √49t 2 ∙ 2t 6 ∙ 7t ∙ √2t 42t√2t

9. DIVISION PROPERTY OF SQUARE ROOT √ 36 49 √ 144 36

10. PROBLEM 5 √ 8x 3 50x 8x 3 = √4  2  x 2  x 50x = √25  2  x 2  x 5 2x 5

11. RATIONALIZING THE DENOMINATOR It’s okay to have a square root in the numerator, but not the denominator. It’s not simplified enough if you keep a square root in the denominator. √3 √7 √3 √7 √21 √49 √21 7 Really equals 1

12. OPERATIONS WITH RADICAL EXPRESSIONS LESSON 10-3

13. COMBINING “LIKE” RADICALS 3√5 and 7√5 have the same radicand. Radicand = number under the square root. -2√9 and 4√3 do not have the same radicand. If two or more numbers have the same radicand, then we can combine them together.

14. PROBLEM 1 What is the simplified form of 2√11 + 5√11? 2√11 + 5√11 We could break it down even more… (√11 + √11) + (√11 + √11 + √11 + √11 + √11) How many √11’s do we have altogether? 7 √11 2√11 + 5√11 = 7 √11

15. WHAT IS THE SIMPLIFIED FORM OF √3 - 5√3? √3 - 5√3 1√3 - 5√3 (1 – 5)√3 -4√3 Got it? 1. 7√2 - 8√22. 5√5 + 6√5

16. PROBLEM 2: WHAT IF THEY DON’T “LOOK LIKE THEY CAN BE SIMPLIFIED? 5√3 - √12 Simplify √12 to see if there is a perfect square. √12 = √4 ∙ 3 = 2√3 So we have 5√3 - 2√3. 5√3 - 2√3 = 3√3

17. GOT IT? 2 1.4 √7 + 2 √28 2.5 √32 - 4 √18

18. PROBLEM 3: USING THE DISTRIBUTIVE PROPERTY 10(6 + 3) = 10(6) + 10(3) = 60 + 30 = 90 In the same way… √10(√6 + 3) Use the Distributive Property (√10 ∙ √6) + (√10 ∙ 3) √60 + 3√10 Break down 60 to find a perfect square. √4 ∙ √15 + 3√10 2 √15 + 3 √10 Can we simplify even more?

19. PROBLEM 3: (√6 - 2 √3)(√6 + √3) (√6 - 2 √3)(√6 + √3) Carefully FOIL (√6)(√6) + (- 2 √3)(√6) + (√6)(√3) + (- 2 √3)(√3) First InsideOutside Last √36 + -2√3 ∙ 6 + √6 ∙ 3 + -2√3 ∙ 3 6 + -2√18 + √18 + -2 ∙ 3 6 – √18 – 6 -√18 = -1 √9 ∙ 2 = -1 ∙ 3 √2 = -3√2

20. GOT IT? 3 1. √2(√6 + 5)

21. GOT IT? 3 2. (√11 – 2) 2

22. GOT IT? 3 3. (√6 – 2 √3)(4 √3 + 3 √6)

23. PROBLEM 4: COJUGATES

24. PROBLEM 4: RATIONALIZING A DENOMINATOR

25. SOLVING RADICAL EQUATIONS LESSON 10-4

26. ESSENTIAL UNDERSTANDING Radical equations = equations with a radicand (square root) Some radical equations can be solved by squaring each side. The expression under the radicand MUST be positive.

27. PROBLEM 1

28. GOT IT? 1

29. PROBLEM 2

30. PROBLEM 3

31. GOT IT? 3

32. EXTRANEOUS SOLUTIONS Take the original equation x = 3. Let’s square each side. x 2 = 9 The solutions would be 3 and -3….right? However, -3 doesn’t fit in our original equation. Sometimes when we square each sides, we create a false solution.

33. n 2 = n + 12 n 2 - n – 12 = 0 (n – 4)(n + 3) = 0 n = 4 and -3 Does both numbers work for n? PROBLEM 4

34. (n – 4)(n + 3) = 0 n = 4 and -3 PROBLEM 4

35. PROBLEM 5 What is the solution √3y + 8 = 2? √3y = -6 Can you ever have a negative as a product of a square root? (√3y) 2 = -6 2 3y = 36 y = 12 Check:

36. LESSON CHECK 1

37. LESSON CHECK 2

38. LESSON CHECK 3

39. LESSON CHECK 4

40. GRAPHING SQUARE ROOT FUNCTIONS LESSON 10-5

41. SQUARE ROOT FUNCTION IS…

42. WHAT IS THE DOMAIN AND RANGE? Domain: What numbers can you put in for x? All positive numbers Range: What kind of numbers are the output of y? All positive numbers

43. PROBLEM 1

44. GOT IT? 1

45. PROBLEM 2

46. GOT IT? 2 When will the current exceed 1.5 amperes?

47. PROBLEM 3

48. PROBLEM 4

49. GOT IT? 3 AND 4

50. LESSON QUIZ

51. HOME WORK #8 – 38 evens On 16 – 24 just make a table. On 30 – 38, tell me the coordinate that the graph will start. You don’t need to graph it.

52. TRIGONOMETRIC RATIOS LESSON 10-6

53. KEY CONCEPT: TRIG RATIOS

55. EXAMPLE 9 12 15 R

56. EXAMPLE 9 12 15 R

57. EXAMPLE 9 12 15 R

58. GOT IT? 1 What is the sine, cosine and tangent of E? 8 15 17 E

59. PROBLEM 2 What is the cosine of 55 degrees? Step 1: Make sure your calculator is in Degree mode. Step 2: Press “cos” and then 55 and then “enter”

60. GOT IT? 2 Use a calculator to compute these trigonometric ratios. 1.Sin 80 2.Tan 45 3.Cos 15 4.Sin 9

61. PROBLEM 3 14 Sin 48 = x x ≈ 10.4

62. GOT IT? 3 To the nearest tenth, what is the value of x in the triangle?

63. FLIP CHART

64. INVERSE OF TRIG RATIOS Sine(Sine -1 ) = 1 Cosine(Cosine -1 ) = 1 Tangent(Tangent -1 ) = 1 Sine and Sine -1 are inverses of each other. Cos(Cos -1 )(45) = 45

65. PROBLEM 4 Find the angle of A in the triangle.

66. GOT IT? 4 In a right triangle, the side opposite angle A is 8mm and the hypotenuse is 12 mm long. What is the angle of A?

67. OUTSIDE ANGLES Angle of Elevation: angle from the horizontal UP to the line of sight. Angle of Depression: angle from the horizontal DOWN to the line of sight.

68. PROBLEM 5 Suppose you are waiting in line for a ride. You see your friend at the top of the ride. How fare are you from the base of the ride?

69. PROBLEM 5

70. HOME WORK #31 – 35, 39 – 42 all