Bonus Section II: Solving Trigonometric Equations

Transcription

1 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II 260 Bonus Section II: Solving Trigonometric Equations (In your text this section is found hiding at the end of 9.6) For what values of x does sin 2 x + cos 2 x = 1? sin 2 x + cos 2 x = 1 is true for sin 2 x + cos 2 x = 1 is called a trigonometric. sin x = 3 is not always a true statement. Our first task is to determine the values of x that 2 make it a true statement. That is what is considered solving the equation. 1. Solve sin x = 3 2 number of solutions. Looking at the graph, we can see that this equation has an infinite We could make a list of all of the solutions: But the common practice is to name all of the solutions by using a parameter. It is unfortunate that WebAssign uses the parameter k. This is NOT the same k we used in 8C to discuss the period of a trigonometric function.

2 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II Solve sin 2 ( x) = 3 4. This is equivalent to solving

3 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II a) Solve cos( 3x) = 1 Remember that the period of cos( 3x) is Start by solving cosu = 1 b) How many solutions are there for cos( x) = 1, where x [ 0, 2π ]? c) Look at the graph above. How many minimum values of cos( 3x) occur on [ 0, 2π ]? d) List the solutions of cos( 3x) = 1, x [ 0, 2π ] by inspecting the graph: e) List the solutions of cos( 3x) = 1, x [ 0, 2π ] by using the parameter

4 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II a) Solve ( sin x) ( tan x) = sin x b) Which of these solutions is on x [ π, π ]

5 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II Solve cos x = sin2x, x [ π, π ]. Since the arguments don t match, use an identity to replace one of the functions.

6 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II Solve 2 sin x sin x 2 +1 = 0, x [ 4π, 4π ].

7 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II Consider the identity sin2x = 2sin x cos x. It could be rewritten as 1 2 sin2x = sin x cos x. a) What is the amplitude of the function f (x) = 1 2 sin2x? b) What is the period? c) Plot f (x) = 1 2 sin2x 1 This is the graph of g(x) = 2 sin2x 2 = ( sin x cos x) 2 = ( sin 2 x) ( cos 2 x) d) For what values of x does (sin 2 x)(cos 2 x) = 1?

8 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II 267 e) For what values of x on [ π, π] does 4(sin 2 x)(cos 2 x) = 1? This is a nice resource: %20TRIG%20EQNS.pdf

9 Fry Texas A&M University Math 150 Spring 2017 Bonus Section II 268 An extra problem for Solving Trigonometric Equations x Determine all of the solutions of the equation sin 2 2 = 1 4

10 Fry Texas A&M University Math 150 Spring 2016 Section Introduction to Functions (The Difference Quotient and Word Problems) We have been studying functions all semester, now it is time to complete this section where the important concept of difference quotient is introduced. Average Rate of Change Assume that the function represented by this graph is a position function. The horizontal axis is time given in seconds. The vertical axis is distance an object has moved from the starting point given in feet. It will be helpful to remember that d = rt so r = What is the position of the object at t = 0? What is the position of the object at t = 4? How far did the object move during those 4 seconds? What is the average speed of the object during those 4 seconds? What is the position of the object at t = 8? How far did the object move during those 8 seconds? What is the average speed of the object during those 8 seconds? How far did the object move between t = 4 and t = 8 What is the average speed of the object during those 4 seconds? Connect the points on the graph for t = 4 and t = 8. What is the slope of that line segment? The slope of the secant line for a position function is

11 Fry Texas A&M University Math 150 Spring 2016 Section How might you go about calculating an estimate for the instantaneous velocity at a point, say at t=4? You might calculate the average velocity between t=4 and t=5. Or better, you might calculate the average velocity between t=4 and t=4.5. Or better yet, you might calculate the average velocity between t=4 and t=4.1 The smaller the interval of time, the closer the average velocity will be to instantaneous velocity. Here we have the graph of some function f (x) Choose a general point (x, f (x)) on the graph. Let h > 0 be some teeny-tiny number. x + h would be just to the right of x. (Imagine that we ve zoomed in on the graph so it looks big, but it s really small.) Plot the point at (x + h, f (x + h)). Connect those two points with a line segment. Calculate the slope of that line segment: This is what is called the. In calculus, you will study what happens as h gets smaller and smaller. In the meantime, we will practice the sometimes-messy algebra that surrounds the Difference Quotient.

12 Fry Texas A&M University Math 150 Spring 2016 Section If f (x) = x 2 +1, simplify the difference quotient: 2. If f (x) = x 1, simplify the difference quotient:

13 Fry Texas A&M University Math 150 Spring 2016 Section If f (x) = x, simplify the difference quotient: Do a job problems I think of them this way: Amount of the job completed = (rate the job is completed) * (time working) Also remember that if it takes someone 3 hours to do a job, then the rate is 1 3 job per hour. 4. Amy can clean the house in 6 hours. Brian can do the same job in 9 hours. In how many hours can they do the job if they work together?

14 Fry Texas A&M University Math 150 Spring 2016 Section Amy and Brian can rake their entire yard in 2 hours when working together. If Brian requires 6 hours to do the job alone, how many hours does Amy need to do the job alone? 6. One pipe can fill a tank 1.25 times faster than a second pipe. When valves to both pipes are opened, they fill the tank in five hours. How long would it take to fill the tank if only the second pipe is used?

15 Fry Texas A&M University Math 150 Spring 2016 Section Writing a Function 7. Suppose there is a piece of sheet metal that is rectangular: 1 foot long and 2 feet wide. Four congruent squares are cut from the corners, so that the resulting piece of metal can be folded and welded into a box. If the length of the sides of the squares is s, write a function that describes the volume of the box in terms of s. 8. Suppose there is a wire 40 inches long. Suppose the wire is cut into 2 pieces, not necessarily equal in length. If each of the pieces of wire is bent to form a square, write a function that describes the sum of the areas of the two squares.

16 Fry Texas A&M University Math 150 Spring 2016 Section Extra Problems for Section 5.1 (Difference Quotient and Word Problems): From Dr. Lynch s Fall 2016 Exam II 1. State the formula for the difference quotient, and then evaluate the difference quotient for the function f (x) = 3x 2 5x. Fully simplify your answer. 2. Someplace to the west of College Station, it takes an orange pipe and a white pipe 3 hours to fill a small swimming pool. It takes 7 hours for the orange pipe to fill the pool by itself. How long would it take the white pipe to fill the pool by itself?

17 Fry Texas A&M University Math 150 Spring 2016 Section From Dr. Scarborough s Math 150 Fall 2015 Exam II. Give the general form of the difference quotient. Apply it to the function f (x) = 3 and then fully simplify. 2 x 4. (One of Erin Fry s exam problems.) Write a function d(x) that outputs the distance between the point (2, 3) and a general point on the graph of f (x) = x 2. (Next semester you can use your function to determine the point on the graph that is closest to (2, 3). )

18 Fry Texas A&M University Math 150 Spring 2017 Section Rectangular Coordinate System (Distance and Circles) A is any set of ordered pairs of real numbers. The of the relation is the set of all first elements of the ordered pairs. The of the relation is the set of all second elements of the ordered pairs. Calculating Distance in the rectangular coordinate system: Find the distance d between the points A(1, 2) and B(-3, 4). Find the distance d between the points C(x 1, y 1 ) and D(x 2, y 2 ) Consider a general point B (x, y) that is 5 units from the origin. This equation is true for all points that are exactly 5 units from the origin. Sketch the graph of this equation.

19 Fry Texas A&M University Math 150 Spring 2017 Section Place the point B at (1, 2). List and graph 4 points that are 3 units from B. Is the origin 3 units from (1, 2)? Write an equation whose solution is all of the points that are 3 units from (1, 2) This is the equation of the centered at with radius We could repeat this procedure starting at an arbitrary point (h, k) and find all the points that are a given distance called r from (h, k). In this case, we would find that This is the standard form of the equation of a centered at with radius. Write the equation of a circle centered at (-3, 0) with radius 2. What is the center of the circle (x + 4) 2 + (y 5) 2 = 6? What is the radius of this circle? Expand this equation and simplify:

20 Fry Texas A&M University Math 150 Spring 2017 Section The General Form of an Equation of a Circle is written Note that there is both an term and a term, but no term. The coefficients on the x 2 term and the y 2 are both Consider the equation 4x 2 + 4y 2 24x + 40y +135 = 0 Is this a circle? If so what is its center? What is its radius? Consider the equation 16x 2 +16y 2 +16x 56y + 5 = 0 Is this a circle? If so what is its center? What is its radius?

21 Fry Texas A&M University Math 150 Spring 2017 Section Extra Problems for Section 4.2 (Circles) From Dr. Lynch s Fall 2016 Exam II 1. Find the center and radius of the circle given by the equation 3x 2 + 3y 2 6x + 4y + 3 = 0 2. Suppose the points (2, -3) and (0, 3) are the endpoints of the diameter of a circle. Find the equation of the circle.

22 Fry Texas A&M University Math 150 Spring 2017 Section Chapter 3B Graphs of Equations (Intercepts and Symmetry of Relations) Intercepts The y-intercept is the point where the graph crosses the. It occurs when The x-intercept is the point where the graph crosses the. It occurs when Find the intercepts of the graph for the following equation: x 2 + xy + 5y 2 = 25. To find the x-intercept To find the y-intercept Find the intercepts of the graph for the following equation: (x 4) 2 + (y + 2) 2 = 9.

23 Fry Texas A&M University Math 150 Spring 2017 Section Symmetry about the x-axis On the figure below, label the vertices in the first and second quadrants. Then reflect the image through the x-axis to create a polygon that is symmetric about the x-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the x- axis iff for each (x, y) on the graph, is also on the graph. We can test an equation to see if its graph will be symmetric about the x-axis. Substitute -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 1. Show that x + y 2 = 4 is symmetric about the x-axis.

24 Fry Texas A&M University Math 150 Spring 2017 Section Symmetry about the y-axis On the figure below, label the vertices in the second and third quadrants. Then reflect the image through the y-axis to create a polygon that is symmetric about the y-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the y- axis iff for each (x, y) on the graph, We can test an equation to see if its graph will be symmetric about the y-axis. Substitute -x for x in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 2. Show that x 4 + y = 9 is symmetric about the y-axis.

25 Fry Texas A&M University Math 150 Spring 2017 Section Symmetry about the origin Informally, a graph is symmetric about the origin if when it is rotated 180 about the origin, the graph looks the same. By definition, a graph is symmetric about the origin iff for each (x, y) on the graph, is also on the graph. Interestingly, if a graph is symmetric about the origin, then for each point on the graph, there is a corresponding point on the graph such that the line segment connecting these two points has the origin as its midpoint. We can test an equation to see if its graph will be symmetric about the origin. Substitute both -x for x and -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 3. Show that x 2 = y is symmetric about the origin.

26 Fry Texas A&M University Math 150 Spring 2017 Section Determine the symmetry of y 3 = 6x 4 x a) Test for symmetry about the x-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about x-axis? b) Test for symmetry about the y-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about y-axis? c) Test for symmetry about the origin by substituting both for and for Equivalent or Not Equivalent? Symmetric or Not Symmetric about origin?

27 Fry Texas A&M University Math 150 Spring 2017 Section Extra Problems for Section 4.3 (Symmetry of Relations) From Dr. Lynch s Fall 16 Exam 2 1. Find the x-intercepts of the graph of the relation given by the equation x 2 x 4 y 6 4 = y 5 2. Determine the symmetry of x 3 y + 3 x + 5y 2 = x y

28 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III 287 Bonus Section III Systems of Non-Linear Equations 1. Plot xy = 1 and x + y = 2 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any. 2. Plot y = x 2 and y = 8 x 2 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

29 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III Plot x 2 + y 2 = 9 and x 2 y = 9 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

30 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III Plot x 2 + y 2 = 17 and x + y = 3 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

31 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III Plot y = 2x 2 + 3x + 4 and y = x 2 + 2x + 3 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

32 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III Plot x 2 + y 2 = 25 and (x 6) 2 + y 2 = 25 on the same graph. Then use algebra to determine the exact point(s) of intersection, if there are any.

33 Fry Texas A&M University Math 150 Spring 2017 Bonus Section III 292 Extra Problem: System of Nonlinear Equations Remember, solving a system of two equations means finding pair(s) of real numbers (x, y) that satisfy all of the equations in the system. Solve (x + 2) 2 + (y + 3) 2 = 9 x 2 + (y + 3) 2 = 3

34 Fry Texas A&M University Math 150 Spring 2017 Vectors 293 Chapter Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional plane. (Think about the Cartesian plane which is drawn by intersecting 2 numbers lines. = 2 ) 3 is the notation for 3-dimensional space. (Imagine another number line passing through the origin, perpendicular to the plane.) Vectors are used to represent quantities such as force and velocity which have both and. The magnitude of a vector corresponds to its. Vectors are directed line segments. (Like the cross between a segment and a ray.) Each of these directed line segments has an initial (starting) point and a terminal (ending) point. Naming Vectors: Vectors in 2 are defined by their terminal point on the Cartesian plane, assuming their initial point is at the origin. For example, the vector 3,4 has its initial point at the origin and its terminal point at ( 3, 4). Notice that distinction between the vector 3,4 and its terminal point ( 3, 4). 3,4 is a vector. It is a directed line segment. It has magnitude and direction. ( 3, 4) is simply an ordered pair of real numbers.

35 Fry Texas A&M University Math 150 Spring 2017 Vectors 294 Definition: A vector v in the Cartesian plane is defined by an ordered pair of real numbers in the form v x, v y. We write v = v x, v y and call v x and v y the of vector v. Specifically v x is the and v y is the of the vector. The graphical representation of 3, 4 is given in Figure 1. Figure 2 Note: Although the vectors in Figure 2 have different initial points and different terminal points, they are all equivalent to 3,4 because they have the same magnitude and direction. Usually what is important about the vector v is not where it is, but how long it is and which way it points. Two or more vectors with the same magnitude and direction are When a vector s initial point is NOT at the origin, we must figure out how to name the vector so that it has the same magnitude and direction as the equivalent vector with initial point at the origin. Note that you can think of the vector 3,4 as set of instructions to get from the initial point to the terminal point: Go to the right 3 and up 4.

36 Fry Texas A&M University Math 150 Spring 2017 Vectors 295 Consider a vector with initial point at (3, -4) and terminal point at (-2, 0). To name this vector think about how you would instruct someone to move from the initial point to the terminal point. Those instructions would be to move right left up down So the name of the vector with initial point at (3, -4) and terminal point at (-2, 0) is What is the name of the vector with initial point at (x 0, y 0 ) and terminal point at (x T, y T )? What is the length of 3,4? Length of a vector: The magnitude or length of a vector v x, v y is denoted v x, v y v x, v y = In 3 the length of a vector v x, v y, v z is denoted v x, v y, v z v x, v y, v z =

37 Fry Texas A&M University Math 150 Spring 2017 Vectors 296 1, 2, 3 = Adding vectors If then u = 3, 5 and u + v = v = 4, 1, Geometrically, imagine picking up vector putting its initial point at the terminal point of Then the new terminal point of terminal point of u + v. v and u. v will be at the Notice that algebraically and geometrically, u + v.= v + u In other words, vector addition is Definition: If u = u x, u y and v = v x, v y, then u + v = In 3 if u = u x, u y, u z and v = v x, v y, v z, then u + v = If 1, 2, 3 + 4, 5, 6 = v = 1, 3, find v + v + v + v = This suggests the reasonableness of the following definition:

38 Fry Texas A&M University Math 150 Spring 2017 Vectors 297 Scalar Multiplication: (When working with vectors, the term scalar refers to a real number.) If u = u x, u y and c is a real number, then the scalar multiple c u = If v = 1, 2, then 2 v = 3 v = v, 2 v and 3 v all have the Notice how same (The WebAssign problems will use the word parallel.) So scalar multiplication is simply the multiplying a vector by a number. The result is another vector that lies on the same A scalar of particular interest is -1. If u = u x, u y, then ( 1) u = u = If v = 1, 2, then v = Notice how v and v lie but point in

39 Fry Texas A&M University Math 150 Spring 2017 Vectors 298 Direction of a vector: We say that two nonzero vectors u and v have the same direction if c opposite directions if c Give two examples of vectors that are not equivalent to 3, 5, but have the same direction. Give two examples of vectors that have the opposite direction of 3, 5. Do 6, 10 and 9, 15 have the same direction? If 6, 10 and 9, 15 have the same direction then These two vectors have the direction :

40 Fry Texas A&M University Math 150 Spring 2017 Vectors 299 Do 4, 9 and 2, 3 have the same direction? If 4, 9 and 2, 3 have the same direction then Unit Vectors A unit vector is a vector with length. Is v = 1, 2 a unit vector? What is the length of 1, 2? 1, 2 = What if you wanted a unit vector that is in the same direction as 1, 2? You would want a scalar c, so that c 1, 2 = c, 2c = 1 c, 2c = Remember that since c 1, 2 is in the same direction as 1, 2, c > 0 so c = Check:

41 Fry Texas A&M University Math 150 Spring 2017 Vectors 300 If v = v x, v y, then is a unit vector in the same direction as v. In other words, to produce a unit vector in the same direction as v, simply divide v by its If v = 1, 3, what is the unit vector that is in the opposite direction as Start by finding a unit vector that is in the same direction as v. v? So is a unit vector in the same direction as Since we want a unit vector in the opposite direction, we will multiply this vector by c = v. is a unit vector in the opposite direction as v. There are two really important unit vectors: i = and j = In 3 i = j = and k = " " Sometimes these are named e 1 = and e 2 = " " In e 3 1 = e 2 " = and e 3 =

42 Fry Texas A&M University Math 150 Spring 2017 Vectors 301 Notice that if v = 3, 5, then If v = v x, v y, then In other words every vector is a of i and j. If v makes a 30 angle with the positive x-axis and then what are the x and y components of v? v = 5 If v makes an angle θ with the positive x-axis, then

43 Fry Texas A&M University Math 150 Spring 2017 Vectors 302 If u = 3, 4 what is the angle θ that x-axis? u makes with the positive If v = 3, 4 what is the angle θ that positive x-axis? v makes with the If w = 3, 4 what is the angle θ that positive x-axis? w makes with the If a = 3, 4 what is the angle θ that positive x-axis? a makes with the

44 Fry Texas A&M University Math 150 Spring 2017 Vectors 303 If u = 5, 2 and v = 1, 5, find the angle θ, between u and v. If u = u x, u y and v = v x, v y, find the angle θ, between u and v.

45 Fry Texas A&M University Math 150 Spring 2017 Vectors 304 Though we have determined the formula for the angle between to vectors, that formula is not the one that is remember and used. Instead, we will use a formula that involves an operation called the dot product. Vector Dot Product If u = u x, u y and v = v x, v y, then the dot product u i v = Notice that the dot product of two vectors is a. The vector dot product is sometimes called the It is important to notice the distinction between the scalar product of two vectors an operation on two vectors which yields a scalar and scalar multiplication of a vector an operation between a scalar and a vector that yields a vector. Notice also near the end of the previous page we had cosθ = u x v x + u y v y u v we could rewrite that as or as u i v =

46 Fry Texas A&M University Math 150 Spring 2017 Vectors 305 Calculate the dot product 3, 4 i 2, 5 = Now determine the angle between the vectors 3, 4 and 2, 5 Calculate the dot product i i j = What is the angle between i and j? cos90 = Notice that since u i v = if u and v are then u i v = In fact u i v = 0 iff Synonyms for perpendicular include In 3 If u = u x, u y, u z and v = v x, v y, v z, then the dot product u i v = 1, 2, 3 4, 5, 6 =

47 Fry Texas A&M University Math 150 Spring 2017 Vectors 306 Find a vector perpendicular to 1, 1 If v x, v y is perpendicular to 1, 1, then How many vectors exist that are perpendicular to 1, 1? How many unit vectors exist that are perpendicular to 1, 1? How do you find a unit vector that is in the same direction as the original vector? You divide the vector by its is perpendicular to 1, 1, and the length of = So is one unit vector perpendicular to 1, 1 and is the other.

48 Fry Texas A&M University Math 150 Spring 2017 Vectors 307 Find a vector If v perpendicular to 3, -4. v is perpendicular to 3, -4, then so v x, v y 3, -4 = Consider choosing v x = because then clearly v y = and so the vector perpendicular to 3, -4 is. Find all the unit vectors perpendicular to 3, -4

49 Fry Texas A&M University Math 150 Spring 2017 Vectors 308 An airplane is flying at 300 miles per hour, heading 30 degrees North of East. What are the magnitudes of the North and East components of the velocity? A wind from due North starts blowing at 40 miles per hour. What is the new velocity of the plane? A river flows at south at 3 mph and a rower rows at 6 mph. What heading should the rower take to go straight east across the river? What if the river flowed at 6 mph and the rower rowed at 3 mph?

50 Fry Texas A&M University Math 150 Spring 2017 Vectors 309 A good dot product problem u = sin Asin Bi + sin Asin B j + cos Ak v = cos 2 Ci + sin 2 C j + cos Bk Determine u i v