LESSON 10-1: THE PYTHAGOREAN THEOREM

SIMPLIFYING RADICALS LESSON 10-2

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

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

GOT IT? 1 Simplify: √72

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

GOT IT? 2 Simplify: -m √80m 9

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

DIVISION PROPERTY OF SQUARE ROOT √ √

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

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

OPERATIONS WITH RADICAL EXPRESSIONS LESSON 10-3

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.

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

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

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

GOT IT? √7 + 2 √ √ √18

PROBLEM 3: USING THE DISTRIBUTIVE PROPERTY 10(6 + 3) = 10(6) + 10(3) = = 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 √ √10 Can we simplify even more?

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 √ √3 ∙ 6 + √6 ∙ √3 ∙ √18 + √ ∙ 3 6 – √18 – 6 -√18 = -1 √9 ∙ 2 = -1 ∙ 3 √2 = -3√2

GOT IT? 3 1. √2(√6 + 5)

GOT IT? 3 2. (√11 – 2) 2

GOT IT? 3 3. (√6 – 2 √3)(4 √3 + 3 √6)

PROBLEM 4: COJUGATES

PROBLEM 4: RATIONALIZING A DENOMINATOR

SOLVING RADICAL EQUATIONS LESSON 10-4

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.

PROBLEM 1

GOT IT? 1

PROBLEM 2

PROBLEM 3

GOT IT? 3

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.

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

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

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 = y = 36 y = 12 Check:

LESSON CHECK 1

LESSON CHECK 2

LESSON CHECK 3

LESSON CHECK 4

GRAPHING SQUARE ROOT FUNCTIONS LESSON 10-5

SQUARE ROOT FUNCTION IS…

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

PROBLEM 1

GOT IT? 1

PROBLEM 2

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

PROBLEM 3

PROBLEM 4

GOT IT? 3 AND 4

LESSON QUIZ

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.

TRIGONOMETRIC RATIOS LESSON 10-6

KEY CONCEPT: TRIG RATIOS

EXAMPLE R

EXAMPLE R

EXAMPLE R

GOT IT? 1 What is the sine, cosine and tangent of E? E

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”

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

PROBLEM 3 14 Sin 48 = x x ≈ 10.4

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

FLIP CHART

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

PROBLEM 4 Find the angle of A in the triangle.

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?

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.

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?

PROBLEM 5

HOME WORK #31 – 35, 39 – 42 all