Chapter 8B - Trigonometric Functions (the first part)

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1 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 79 Chapter 8B - Trigonometric Functions (the first part) Recall from geometry that if 2 corresponding triangles have 2 angles of equal measure, then they are. E C A D B Since ABC and ADE are both right triangles sharing the common angle A, they are similar triangles. When 2 triangles are similar, it means the lengths of their corresponding sides are proportional. So AD AB = adjacent leg small adjacent leg big = DE BC opposite leg small = opposite leg big = AE AC hypotenuse small = hypotenuse big If we carefully use algebra to rearrange AD AB = DE BC, we see that BC AB = DE AD or rather that and similarly, since AE AC = DE BC BC then it must be true that AC = DE AE which stated more plainly says

2 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 80 These rather amazing facts allow us to say that given a right triangle with an angle measured θ, the following ratios are constant (no matter how large or small the triangles are!). sinθ =!!!! cosθ = tanθ =!!!! cotθ = cscθ =!!!!! secθ = Tom s Old Aunt!! Sat On Her!!! Coffin And Howled Notice that sinθ cosθ!!!!!!! In other words sinθ cosθ Similarly cosθ sinθ =!!!! 1 cosθ = In other words 1 cosθ =!!!! Similarly 1 sinθ =

3 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 81 Consider the equilateral triangle sin π 3 = cos π 3 = tan π 3 = csc π 3 = sec π 3 = π cot = 3 sin π 6 = cos π 6 = tan π 6 = csc π 6 = sec π 6 = π cot = 6

4 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-I! Page 82 Consider the isosceles right triangle sin π 4 = cos π 4 = tan π 4 = csc π 4 = sec π 4 = π cot = 4 The trouble with these definitions of our trigonometric functions is that they are defined only for 0 < θ < π. Later we will extend the definitions of these functions for any real value of θ. 2

5 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 83 Chapter 5A Polynomials Polynomial Degree Leading Term Constant Term f (x) = 4x 3 + 2x 5 g(x) = 17x 5 + 6x 3 h(x) = πx 6 17 c(x) = 14 p(x) = p n x n + p n 1 x n 1 +!+ p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers, the p i 's are real numbers and p n 0 r(x) = 2 + 3x 5x 4 We ve already graphed polynomials of degree 1 (lines) Now we will pay special attention to when the graph is either above or below the x-axis. f (x) = x + 3 g(x) = 4 2x Where does x + 3 = 0? At x = When x < -3 x+3 < 0 so the graph is the x-axis. When x > -3 x+3 > 0 so the graph is the x-axis. x+3 changes sign at x=-3. x+3 goes from negative to positive. Where does 4 2x = 0? At x = When x < 2 4-2x > 0 so the graph is the x-axis. When x >2 4-2x < 0 so the graph is the x-axis. 4-2x changes sign at x=2. 4-2x goes from positive to negative.

6 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 84 f (x) = ( x + 3) ( x 2) g(x) = 2(2 x)(x +1) Where does ( x + 3) ( x 2) = 0? At x = and x = These zeros divide the x-axis into 3 regions. Where does g(x) = 2(2 x)(x +1)? At x = and x = These zeros divide the x-axis into 3 regions. Notice that when x < -3, both factors are negative. x+3 < 0 and x-2 < 0. If both factors are negative, their product will be positive. See how the graph is above the x-axis! Notice that when x < -1, the factors have different signs. x+1 < 0 but 2-x > 0. That means their product will be negative. See how the graph is below the x-axis. When -3 < x < 2, the factors have different signs. x+3 > 0 but x-2 < 0. That means their product will be negative. See how the graph is below the x-axis When x > 2, both factors are positive x+3 > 0 and x-2 > 0. So product of these two factors is positive. Again the graph is above the x-axis! When -1 < x < 2, the factors are both positive. x+1 > 0 and 2-x > 0. If both factors are positive, their product will be positive and the graph will be above the x-axis. When x > 2, the factors again have different signs. x+1 > 0 but 2-x < 0. Since one factor is positive and the other factor is negative, their product will be negative and the graph of g(x) will be below the x-axis.

7 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 85 Now we are ready for cubics -- polynomials of degree 3. The first two are easy. They are really graphs from 4C. Cubics: f (x) = x 3!!!!!!!! g(x) = (x 2) 3 Leading Term: Constant Term: y-intercept: Leading Term: Constant Term: y-intercept: As x, x 3 As x, (x 2) 3 As x, x 3 As x, (x 2) 3 Roots: Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 0, x 3 < 0 When x > 0 x 3 > 0 Roots: Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 2, (x 2) 3 > 0 When x > 0 (x 2) 3 < 0

8 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 86 h(x) = 2(x + 3)(x 1) 2 Leading Term: Constant Term: So the y-intercept is As x, 2(x + 3)(x 1) 2 As x, 2(x + 3)(x 1) 2 Roots: Now we would like to solve 2(x + 3)(x 1) 2 > 0 (because when 2(x + 3)(x 1) 2 > 0, the graph of h(x) will be the x-axis.) Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product h(x) > 0 when so the graph of h(x) is above the x-axis when

9 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 87 p(x) = 1 4 (x + 2)(x 2)(3x 4) Leading Term: Constant Term: So the y-intercept is As x, 1 4 (x + 2)(x 2)(3x 4) As x, 1 4 (x + 2)(x 2)(3x 4) Roots: Solve p(x) > 0. Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product p(x) > 0 when so the graph of p(x) is above the x-axis when

10 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 88 True or False: Some cubics never intersect the x-axis. Some cubics intersect the x-axis in exactly one place. Some cubics intersect the x-axis in exactly two places. Some cubics intersect the x-axis in three places. Some cubics intersect the x-axis in four places. Quartics f (x) = x 4 +1!!!!!!! g(x) = 1 4 (x 2) 4 Leading Term: Leading Term: Constant Term: Roots: As x, x 4 +1 As x, x 4 +1 Constant Term: Roots: As x, 1 (x 2) 4 4 As x, 1 (x 2) 4 4

11 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 89 h(x) = x 3 (2x + 7) Leading Term: Constant Term: So the y-intercept is As x, x 3 (2x + 7) As x, x 3 (2x + 7) Roots: Solve h(x) = x 3 (2x + 7) > 0 p(x) = (x 2 1)(x + 2) 2 Leading Term: Constant Term: So the y-intercept is As x, p(x) = (x 2 1)(x + 2) 2 As x, p(x) = (x 2 1)(x + 2) 2 Roots: Solve p(x) = (x 2 1)(x + 2) 2 > 0

12 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 90 Polynomials of Higher Degree In general, to understand the behavior of a polynomial, 1. Plot the y-intercept 2. Determine the behavior of the polynomial for large positive values of x and for large negative values of x. The behavior of the polynomial at these extremes will be dominated by the leading term, the term with the highest power of x. 3. Find the zeros. 4. Determine where the polynomial is positive and negative because this will tell you where the graph is above and below the x-axis. f (x) = x 5!!!!!!! g(x) = x 5!!! Leading Term: Constant Term: Leading Term: Constant Term: As x, x 5 As x, x 5 As x, x 5 As x, x 5 Roots: Roots: x 5 > 0 x 5 > 0

13 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 91 h(x) = (3x 1) 2 (x 2)(x +1)(x + 3)!!!!!!! Leading Term: Constant Term: So the y-intercept is As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) Roots: Solve h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) > 0

14 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 92 f (x) = x(x 3) 2 (2x + 5) 2 Degree of the polynomial: Leading Term: Constant Term: y-intercept: As x, x(x 3) 2 (2x + 5) 2 As x, x(x 3) 2 (2x + 5) 2 Zeros: Solve: x(x 3) 2 (2x + 5) 2 > 0

15 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 93 True or False: All linear functions cross the x-axis. True or False: All quadratic functions cross the x-axis. True or False: All cubic functions cross the x-axis. True or False: All quartic functions cross the x-axis. True or False: All quintic functions cross the x-axis. True or False: All polynomials functions of even degree cross the x-axis. True or False: All polynomials functions of an odd degree cross the x-axis. Extra Problems: Text:! 1-8!

16 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page 94 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values where the denominator is 1. g(x) = 1 x +1 a) domain b) y -intercept! c) x -intercept A fraction is zero when its is zero and its is NOT zero. That is why the graph of 1 x +1 never the d) vertical asymptote Vertical asymptotes occur where the is zero, but the is not zero. e) Solve g(x) > 0 f) for large x, g(x) acts like, (This is the quotient of the.) so as x, g(x) --> and as x g(x) -->

17 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page h(x) = x x +1 a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) Solve h(x) > 0 f) for large x, (think about estimating) h(x) acts like, (This is the quotient of the ) so as x, h(x) --> and as x h(x) -->

18 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page p(x) = x2 +1 x +1 a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) Solve p(x) > 0 f) For large x, (think about estimating) p(x) acts like, (This is the quotient of the ) so as x, p(x) --> and as x p(x) -->

19 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page 97 A hole in the graph indicates a place where the function is undefined, but the function s behavior is not asymptotic. Understanding Holes in Graphs: 4. f (x) = x2 1 x +1! a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) there is a hole at f) the y - coordinate of hole g) Solve f (x) > 0 h) For large x, f (x) acts like, so as x, f (x) --> and as x f (x) -->

20 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page g(x) = x2 1 x 2 +1! a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) there is a hole at f) the y - coordinate of hole g) Solve g(x) > 0 h) For large x, g(x) acts like, so as x, g(x) --> and as x g(x) -->

21 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page 99 x h(x) = 3( x 2 9)(x 1)! a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) there is a hole at f) the y - coordinate of hole g) Solve h(x) > 0 h) for large x, h(x) acts like, so as x, h(x) --> and as x h(x) -->

22 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page p(x) = 2x 3 + 6x 2 x 3 + 3x 2 4x 12 a) domain b) y -intercept! c) x -intercept d) vertical asymptote e) there is a hole at f) the y - coordinate of hole g) Solve p(x) > 0 h) For large x, p(x) acts like, so as x, p(x) --> and as x p(x) -->

23 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page 101 To graph a rational function, a) Evaluate the function at x = 0, this is the b) Find the values for x for which the numerator is zero, but the denominator is not zero. This is where the graph c) Find the values for x for which the denominator is zero, but the numerator is not zero. This is where the graph d) Find the values of x for which both the numerator and the denominator are zero. This is where there is e) To find the y - coordinate of the hole: If there is a hole at x = a, then ( x a) is a factor of both the numerator and the denominator. The rational function f (x) can be written in the form (x a) f (x) = p(x) ( x a) q(x). It could be that p(x) = 1 and/or q(x) = 1.! Let f (x) = p(x) q(x), then the y - coordinate of the hole is f a ( ). f) Simplify the quotient of the leading terms of the numerator and the denominator. The end behavior of this function is the same as the end behavior of the given function. g) Determine where the function is greater than 0.! This is where the graph of the function is.

24 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page 102 The Generalized Technique for Determining End Behavior of Rational Functions: 8. Use the Generalized Technique for Determining End Behavior to determine the end behavior of f (x) = x x 4 Step 1: Determine the highest power of x involved in the function. In this case it is. Step 2: Multiply the rational function by 1 in the special form: Step 3: Simplify each term. Step 4: Examine the end behavior of each term. Step 5: Use this information to determine the end behavior of the rational function. As x, f (x). So the graph of f (x) has a horizontal asymptote of

25 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5B! Page Use the Generalized Technique for Determining End Behavior to determine the end behavior of g(x) = 5x x 3 Step 1: Determine the highest power of x involved in the function. In this case it is. Step 2: Multiply the rational function by 1 in the special form: Step 3: Simplify each term. Step 4: Examine the end behavior of each term. Step 5: Use this information to determine the end behavior of the rational function. As x, g(x). So the graph of g(x) has a horizontal asymptote of Extra Problems:! Text:! 1-3, 5-26!!

26 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1D! Page 104 Chapter 1D - Rational Expressions A rational expression is the quotient of two Simplifying rational expressions is like reducing rational numbers: = = 5 6 Notice that = 3+12, but we would not cancel the 12 s! When simplifying rational expressions we look for common in the numerator and denominator. In the previous section, the rational functions were already factored. Sometimes they are found in a messier form. In this section we ll practice the algebraic skills necessary to write a given rational function as the quotient of two factored polynomials. 1. f (x) = 2 x x 3 Domain

27 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1D! Page g(x) = x 4 x +1 x2 8x +16 x 2 1 Domain 3. h(x) = 8 x + 2 i x 2 x 2 x 2 1 Domain

28 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1D! Page f (x) = x2 4x 21 (x 2 + 7x +12) x + 4 Domain 5. g(x) = x 3 1 x +1 x 1!!! Domain x 2 + 2x +1

29 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1D! Page 107 A compound or complex fraction is an expression containing fractions within the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division. 6. h(x) = 3 x + 1 2x 7 5 x +1 Begin by getting a common denominator in the numerator and a common denominator in the denominator.

30 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1D! Page 108 This is a function of 2 variables. We won t try to graph it, we ll just simplify it. 7. f (x, h) = 3 ( x + h) 3 2 x 2 h Note: In the traditional presentation of Math 150, students learn to simplify rational expressions before the concepts of function and domain are defined. So in some WebAssign problems for 1D, you ll be asked for the restrictions on x. You will want to report the values of x that make the function undefined even before it is simplified. Extra Problems: Text: 1-25

31 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-II! Page 109 Chapter 8B - Trigonometric Functions (the second part) Another way to define the trigonometric functions: Given a circle of radius one, (often called the unit circle) with a radial line drawn at an angle θ, measured counterclockwise from the positive x-axis, the radial line intersects the circle at a point (x, y). The trigonometric functions can then be defined as sinθ cosθ tanθ cscθ secθ cotθ Find cos 3π 4

32 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-II! Page 110 Let θ be an angle in standard position. The reference angle θ is the acute angle formed by the terminal side of θ and the x -axis. cos 5π 6 sin 4π 3

33 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-II! Page 111 sin π 6 tan 2π 3 Reference Angle Theorem: Let trig( θ ) be any one of the six trigonometic functions defined above (on page 26).!!!! Then trig( θ ) = ±trig( θ )!!!! The correct sign is determined by the quadrant of θ.

34 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 8B-II! Page 112 tan 7π 6 csc 5π 3

35 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 113 Chapter 2A Solving Equations The solution of an equation is a value (or a set of values) that yields a true statement (an identity) when substituted for the variable in an equation. The word solve means determine Solving Linear Equations Linear equations are equations involving only polynomials of degree one. Examples include 2x +1 = 4 and 4x + 2 = x 3 The algebraic techniques to find the solutions to these equations are simple, but I want you to keep in mind that there is a geometric interpretation associated with the equation. We are looking for the intersection of two linear functions. Here are the graphs of!!!!!! Here are the graphs of and!!!!!! and!!!!!!! See how the lines intersect at!!!!! See how the lines intersect at x =!!!!!!! x = See how substituting x =!!!! See how substituting x = into 2x +1 = 4!!!!!!!! into 4x + 2 = x 3 makes the statement true.!!!!!! makes the statement true.

36 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 114 Solving Quadratic Equations Quadratic equations are equations involving only polynomials of degree two. Examples include x 2 x 6 = 2x 2! and! 5x 2 5x = 2x 2 3x +1 Geometrically, such equations could represent the intersection of a parabola and a line or the intersection of two parabolas. x 2 x 6 = 2x 2 5x 2 5x = 2x 2 3x x 2 = x 1

37 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 115 There are several algebraic techniques for solving quadratic equations. First there are two important ideas to understand: If ab = 1, must either a = 1 or b = 1? If ab = 0, must either a = 0 or b = 0? Solve Quadratic Equations by Factoring 1. Solve Put in standard form This is called the Zero-Product Principle. a) 1 2x 2 = x Factor Set each factor equal to zero Put in standard form b) 6x 2 = x +15 Factor Set each factor equal to zero

38 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 116 Solve by completing the square. 2. Solve the following quadratic equation by completing the square: 2x 2 + x 8 = 0 If ax 2 + bx + c = 0, then x = This is called 3. Solve using the quadratic formula: 2x 2 = 6x 3

39 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 117 Solving equations in quadratic form 4. Solve a) x 4 5x 2 6 = 0 b) x x 5 +1 = 0 c) 4x 12 9x = 0 d) x x = 0

40 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 118 Solving Rational Equations Rational equations are equations involving rational functions. Solving rational equations is equivalent to finding the intersection(s) between 2 rational functions. Algebraically we set one side of the equation equal to zero and remember that a fraction is zero when 5. Solve a) 1 2x = x

41 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 119 b) 4 x 4 3 x 1 = 1

42 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 120 Radical Equations To Solve Radical equations: I. Isolate one radical. II. Raise both sides of the equation to the appropriate power to remove the radical.! (Usually this means square both sides of the equation.) III. Repeat the process until all radicals have been removed IV. Check for extraneous solutions! REMEMBER: (a + b) 2 a 2 + b 2! (a + b) 2 = 6. Solve a) 3 = x + 2x 3

43 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 121 b) 8x +17 2x + 8 = 3 c) x + x 5 = 1

44 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 122 Equations with Absolute Values!! Recall the definition of x = 7. Solve a) x = 4 b) x 3 = 2 c) 8 x + 2 = 7

45 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 123 d) x = 2 e) x 2 x 12 = 8!!

46 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page 124 Equations with several variables Equations in science and engineering often include many variables, and it is useful to be able to solve for one of the variables in terms of the others.!! Warm up: If 1 x = , then what is the value of x? 3 8. The following equation comes from the physics of circuits 1 R eq = 1 R R R 3 Solve for R eq.

47 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2A! Page T s = 2π m k Solve for k. 10. Solve for y x = 3y 2y 5 Extra Problems: Text: 7-56

48 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 126 Chapter 2B - Solving Inequalities True or False: 4 < 4!!! True or False: 4 4 Also though 3 < 4, -4 < -3 which could also be written -3 > -4. Written more generally, if a < b, then This leads to the rule that when you multiply both sides of an inequality by a negative number, you change the direction of the inequality. 1. Solve a) 1 x > x + 3 b) 1 2x < 17 4x 8 x

49 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 127 c) 2x + 4 < x + 2 < x d) 4x x 3 x

50 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 128 Absolute Value Inequalities Determine the solution sets for the following inequalities Number Line Inequality Interval Notation x 3 x > 3 x > 3 x < 3 2. Solve a) x < 4 b) x 2

51 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 129 c) x + 2 < 1 d) 2x 4 3 e) 5 2x > 4

52 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 130 f) 4 x 3 > 1

53 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 131 Nonlinear Inequalities Consider a,b! ab > 0, then either or If ab < 0 then either or To solve nonlinear inequalities: I. Move every term to one side (make one side zero). II. Factor the nonzero side. III. Find the critical values. (Critical values make the expression zero or undefined.) IV. Let the critical numbers divide the number line into intervals. V. Determine the sign of each factor in each interval. VI. Use the sign of each factor to determine the sign of the entire product or quotient. 3. Solve a) x 2 x 6

54 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 132 TRUE or FALSE: 2 < 3 TRUE or FALSE: 2x < 3x because when x < 0, So don t multiply both sides of an inequality by a variable that could be either positive or negative. c) 2 x < 1

55 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 2B! Page 133 d) 2x + 3 6x +1 > x x Extra Problems: Text: 1-34

56 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1E! Page 134 Chapter 1E - Complex Numbers 16 exists! So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers. And x!, x 2 0 But there exists a number (that is not an element of! ) named i and i 2 = Since i 2 =, 1 =, so 16 = i is not used in ordinary life, and humankind existed for 1000 s of years without considering i. i is, however, a legitimate number. i, products of i, and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was termed imaginary! i and numbers like 4i and 3i are called Numbers like 2 + i and i are called More formally! is the set of all numbers When a complex number has been simplified into this form, it is called a is called the part. b is called the real part. So for the complex number 4 3i, is the real part while is the imaginary part. 1. Put the following complex numbers into standard form: a) ( ) + ( ) = b) ( 1+ 2i) (3 4i) = c) ( 5 + 4i)(3 2i) = d) i 3 = e) i 4 = f) i 5 =

57 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1E! Page 135 Graphing Complex Numbers When we graph elements of!, we use When we graph elements of!, we use the complex plane which will seem a lot like the Cartesian plane. In the Cartesian plane, a point represents In the complex plane, a point represents The horizontal axis is the The vertical axis is the 2. Graph and label the following points on the complex plane: A 3+ 4i!!! B 2 + i C 3 2 3i!!! D 3i E 4!!! F 1 2i Absolute Value of a Complex Number: If z! then z is defined as its Calculate 3+ 4i (How far is 3+ 4i from the origin?) 3+ 4i =

58 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1E! Page 136 Consider a + bi, an arbitrary element of!.!! What is a + bi? (How far is a + bi from the origin?) In general z = Notice that z is a real number. It is z s distance from the origin. We did not use i to calculate z. 3. Calculate 5 + i. (How far is 5 + i from the origin?) 5 + i = Complex Conjugate If z = 3+ 4i, then z =. If z = 1 2i, then z =. Graphically the complex conjugate is the of the number through the More generally if z = a + bi, then z =

59 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1E! Page Put the following complex numbers into standard form. a) 2 3i =!!! b) 4i = c) 5 =!!! d) 3i 2 = Finally notice that ( 3 + 4i) ( 3 4i) = More generally ( a + bi) ( a bi) = in other words ziz = Distance Between Two Complex Numbers Plot and label two points in the complex plane z 1 = 3+ 5i and z 2 = 1+ 2i The distance between z 1 and z 2 is Just for fun, calculate z 1 z 2 = And z 1 z 2 = What we have seen is that for this particular z 1 and z 2, the distance between these points is equal to. But this is actually true for all complex numbers. The distance between two general points z 1 and z 2 is

60 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 1E! Page Solve a) x 2 + 2x + 2 = 0 Remember if m and n are zeros of the function f (x) = ax 2 + bx + c, then f (x) can be rewritten as f (x) = How could that be applied here? Could we factor x 2 + 2x + 2 over the complex numbers? x 2 + 2x + 2 = b) 1 2 x 2 3x + 7 = 0 Extra Problems: Text: 1-12

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