Find the component form and magnitude of the vector where P = (-3,4), Q = (-5, 2), R = (-1, 3) and S = (4, 7)

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1 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.1: Vectors in the Plane What you'll Learn About Two Dimensional Vectors/Vector Operations/Unit Vectors Direction Angle/Applications of Vectors Find the component form and magnitude of the vector where P = (-3,4), Q = (-5, 2), R = (-1, 3) and S = (4, 7) A) PQ B) RS C) 3 QS D) 2 QR PS 1 P a g e

2 Let u = 1, 3, v = 4, 7 and w = 2, 5 vector.. Find the component form of the A) u + v B) u w C) 2u + 3w D) -2u 3v 2 P a g e

3 Find a unit vector in the direction of the given vector. Write your answer in component form and as a linear combination of the standard unit vectors i and j A) u = 1, 3 B) v = 4, 7 w = 2, 5. 3 P a g e

4 Find the component form of the vector v with the given magnitude and angle. A) v B) v Find the magnitude and direction angle of the vector. A) 6, 8 B) 6i 8j C) 10cos235 i sin235 j 4 P a g e

5 Navigation A) An airplane is flying on a bearing of 135 at 435 mph. Find the component form of the velocity of the airplane. B) An airplane is flying on a compass heading(bearing) of is blowing with the bearing 220 at 30 mph. - Find the component form of the velocity of the airpane. 315 at 300 mph. A wind - Find the component form of the velocity of the wind. - Find the actual ground speed and direction of the airplane 5 P a g e

6 C) A ship is heading due south at 15 mph. The current is flowing northwest at 3 mph. Find the actual bearing and speed of the ship. Shooting a basketball: A basketball is shot at an angle 12m/sec. a. Find the component form of the initial velocity. 65 with an initial speed of b. Give an interpretation of the horizontal and vertical components of the velocity. Combining Forces: A force of 40 lbs acts on an object at angle of 20. A second force of 65 pounds acts on the object at an angle if 25. Find the direction and magnitude of the resultant force. 6 P a g e

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8 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.2: Dot Product of Vectors What you'll Learn About The Dot Product/Angle Between Vectors Projecting One Vector onto another/work Find the dot product of of u and v. A) u = 1, 3, v = 4, 7 B) u = 3, 5, v = 2, 6 C) (2i j) (3i 5j) Find the angle between the two vectors. A) u = 1, 3, v = 4, 7 A) u = 1, 3, v = 4, 7 8 P a g e

9 Find the angle between the two vectors. B) u = 3, 5, v = 2, 6 Determine if the vectors are parallel, orthogonal, or neither. A) u = 2, 3, v = 6, 4 B) u = 3, 5, v = 2, 6 C) u = 2, 10, v = 1, 5 D) u = 2, 10, v = 1, 5 9 P a g e

10 Given u = 6, 2, v =, 5 5. Using vectors, trigonometry, and geometry, find the coordinate on the vector v that will make a perpendicular segment from vector u to vector v. 10 P a g e

11 Tim is setting on a sled on the side of a hill inclined at 45 degrees. The combined weight of Tim and the sled is 140 pounds. Tony is using a rope tied to the sled to keep it from sliding down the hill. a) Use a vector to represent the force due to gravity. b) Use a vector to represent the side of the hill. c) What force (magnitude) is required for Tony to keep the sled from sliding down the hill? 11 P a g e

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13 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.3: Parametric Equations and Motion What you'll Learn About Parametric Equations/Parametric Curves/Eliminating the Parameter Lines and Line Segments/Simulating Motion with a grapher A ship is heading northwest at 12 mph. a. Find the component form of the ship after 1 hour. b. Find the component form of the ship after 2 hours. c. Find the component form of the ship after 3 hours. d. Write the parametric equations for the boat at any time t. 13 P a g e

14 For the given parameter interval, graph the parametric equations A. x = t 2 2 y = 3t on the interval 0 t 3 B. x = t 2 2 y = 3t on the interval 0 t 5 C. x = t 2 2 y = 3t on the interval 3 t 1 D. x = t 2 2 y = 3t on the interval t 14 P a g e

15 Eliminate the parameter and identify the graph of the parametric curve A) x = t 2 2 y = 3t on the interval t B) x = 1 2t y = 2 t C) x = 2cos(t) y = 2sin(t) 0 t 2 D) x = 2cos(t) y = 2sin(t) 0 t 2 15 P a g e

16 Find a parametrizations for the curve. A) The line through the points A = (-2, 3) and B = (3, 6) B) The line segment with endpoints A = (-1, 5) and B = (2, 11) 16 P a g e

17 37. Ben can sprint at the rate of 20ft/sec. Jerry sprints at 15ft/sec. Ben gives Jerry a 15ft head start. The following parametric equations can be used to model a race. Ben: x 1 = 20t 15 y 1 = 2 Jerry: x 1 = 15t y 1 = 4 a) Find a viewing window to simulate a 100 yard dash. Graph simultaneously with t starting at t = 0 and Tstep =.05. b) Who is ahead after 2 seconds? 3seconds? 4 seconds? Famine Relief Air Drop: A relief agency drops food containers form an airplane on a war torn famine area. The drop was made from an altitude of 2000 feet above ground level. A) Use an equation to model the height of the containers (during free fall) as a function of time t. b) Use parametric mode on your calculator to simulate the drop during the first 6 seconds. c) After 5 seconds of free fall, parachutes open. How many feet above the ground are the food containers when the parachutes open? 17 P a g e

18 Height of a pop-up: A baseball is hit straight up form a height of 4 feet with an initial velocity of 70 ft/sec. a) Write an equation that models the height of the ball as a function of time. b) Use parametric mode to simulate the pop-up. c) Use parametric mode to graph height against time. (Let x(t) = t) d) How high is the ball after 3 seconds? e) What is the maximum height of the ball? How many seconds does it take to reach its maximum height? Hitting a baseball: Kevin hits a baseball at 3 feet above the ground with an initial speed of 150ft/sec at angle of 18 degrees with the horizontal. Will the ball clear a 20 feet wall that is 400 feet away? 18 P a g e

19 The men s horseshoe pitching court has metal stakes 40 feet apart. The stakes stand 18 inches out of the ground. a. Alan pitches a horseshoe at 45 feet per second, at a 14 angle to the ground. He releases the horseshoe at about 3 feet above the ground and 2 feet in front of the stake at one end. Write parametric equations modeling a typical throw. b. How long is the thrown horseshoe in the air? c. How close to 40ft is the horizontal component when the horseshoe hits the ground? 19 P a g e

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21 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.4: Polar Coordinates What you'll Learn About Polar Coordinate System/Coordinate Conversion Equation Conversion/Finding Distance using Polar coordinates Plot the given polar coordinate and then give the other 3 representations of the polar coordinate. A) 2,30 B) 3 3, 4 21 P a g e

22 Given the polar coordinates of a point, find its rectangular coordinates. A) 2,30 B) 3 3, 4 22 P a g e

23 Without graphing, use an algebraic method to convert the following polar coordinates to rectangular. 5 A), 6 3 B) 2, 200 Given the rectangular coordinates, find 4 polar coordinates that represent the coordinate. A) P(-1, 1) B) (5, - 6) C) (-3, 0) 23 P a g e

24 Convert the polar equation to rectangular form and identify the graph. A) r 4cos B) r 4sec C) r cot 4 D) r 2cos 6sin 24 P a g e

25 Convert the rectangular equation to polar form. Sketch the graph of the rectangular equation. A) x = 3 B) y = 3 B) 5x 10y = 20 C) (x - 3) 2 + (y 2) 2 = P a g e

26 Using a radar tracking system: Radar detects two planes at the same altitude. Their polar coordinates are (8 miles, 110 ) and (5 miles, How far apart are the airplanes? 15 ). 26 P a g e

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28 PRE-CALCULUS: by Finney,Demana,Watts and Kennedy Chapter 6: Applications of Trigonometry 6.5: Graphs of Polar Equations What you'll Learn About Polar Curves and Parametric Curves/Symmetry/Analyzing Polar Graphs Rose Curves/Limacon Curves/Other Polar Curves Graph the following polar curve A) r 5 B) 6 28 P a g e

29 A) Graph the following: y = 4sinx Graph the following polar curve: r 4sin B) r 2sin 29 P a g e

30 C) Graph the following: y = -2sinx Graph the following polar curve: r 2sin D) r 4sin 30 P a g e

31 A) Graph the following: y = 4cosx Graph the following polar curve: r 4cos B) r 2cos 31 P a g e

32 C) Graph the following: y = -2cosx Graph the following polar curve: r 2cos D) r 4cos 32 P a g e

33 Summary of Polar Circles Circles always move counterclockwise The circle completes itself from 0 to π. Going from 0 to 2π would retrace the original circle The number in front of sine is the radius Equations with positive sine start at the origin and then move counterclockwise up Equations with negative sine start at the origin and then move counterclockwise down Equations with positive cosine start at the radius on the positive side of the pole and then move counterclockwise up and back toward the origin Equations with negative cosine start at the radius on the negative side of the pole and then move counterclockwise down and back toward the origin 33 P a g e

34 Graph the following polar curve: r 2 2cos r 33cos 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 2. What is the value of when 2. What is the value of when r = 2? r = 3? Graph the following polar curve: r 2 2cos r 3 3cos 34 P a g e

35 Graph the following polar curve: r 2 2cos r 3 3cos 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 2. What is the value of when 2. What is the value of when r = 2? r = 3? Graph the following polar curve: r 2 2cos r 33cos 35 P a g e

36 Graph the following polar curve: r 2 2sin r 33sin 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 2. What is the value of when 2. What is the value of when r = 2? r = 3? Graph the following polar curve: r 2 2sin r 3 3sin 36 P a g e

37 Graph the following polar curve: r 2 2sin r 3 3sin 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 2. What is the value of when 2. What is the value of when r = 2? r = 3? Graph the following polar curve: r 2 2sin r 33sin 37 P a g e

38 Summary of the Cardiod: r a bcos For a polar equation to be considered a cardiod, the values of a and b must be the same. Plug in 0. This will give you the value of r and where you will start moving counterclockwise. The value of a will tell you where the curve is at on the y-axis 3 (when and ) 2 2 To complete the entire shape 0 2 The value of a b will tell you how far out on the x-axis the curve is If b is negative the curve will be on the left side of the pole If b is positive the curve will be on the right side of the pole Summary of the Cardiod: r a bsin For a polar equation to be considered a cardiod, the values of a and b must be the same. Plug in 0. This will give you the value of r and where you will start moving counterclockwise. The value of a will tell you where the curve is at on the x-axis (when 0 and ) To complete the entire shape 0 2 The value of a b will tell you how far out on the y-axis the curve is If b is negative the curve will be below the pole If b is positive the curve will be above the pole 38 P a g e

39 Graph the following polar curve: r 2 3cos r 2 4cos 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 3. What is the value of when 3. What is the value of when?? 2 2 Graph the following polar curve: r 2 3cos r 2 4cos 39 P a g e

40 Graph the following polar curve: r 2 3cos r 2 4cos 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 3. What is the value of when 3. What is the value of when?? 2 2 Graph the following polar curve: r 2 3cos r 2 4cos 40 P a g e

41 Graph the following polar curve: r 2 3sin r 2 4sin 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 3. What is the value of when 3. What is the value of when?? 2 2 Graph the following polar curve: r 2 3sin r 2 4sin 41 P a g e

42 Graph the following polar curve: r 2 3sin r 2 4sin 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of when 2. What is the value of when r = 0? r = 0? 3. What is the value of when 3. What is the value of when?? 2 2 Graph the following polar curve: r 2 3sin r 2 4sin 42 P a g e

43 Summary of the Limacon: r a bcos For a polar equation to be considered a Limacon with a loop, the value a must be smaller than b. Plug in 0. This will give you the value of r and where you will start the curve moving counterclockwise back to the pole. The value of a will tell you where the curve is at on the y-axis 3 (when and ) 2 2 To complete the entire shape 0 2. The value of a b will tell you how far out on the x-axis the curve is The value of a b will tell you how far out on the x-axis the loop is If b is negative the curve and the loop will be on the left side of the pole If b is positive the curve and the loop will be on the right side of the pole Summary of the Limacon: r a bsin For a polar equation to be considered a Limacon with a loop, the value a must be smaller than b. Plug in 0. This will give you the value of r and where you will start the curve moving counterclockwise back to the pole. The value of a will tell you where the curve is at on the x-axis (when 0 and ) To complete the entire shape 0 2. The value of a b will tell you how far out on the y-axis the curve is The value of a b will tell you how far out on the y-axis the loop is If b is negative the curve and the loop will be below the pole If b is positive the curve and the loop will be above the pole 43 P a g e

44 Graph the following polar curve: r 3 2sin r 3 2cos 1. What is the value of r when 1. What is the value of r when 0? 0? 2. What is the value of r when 2. What is the value of r when?? 2. What is the value of when 2. What is the value of when r = 3? r = 3? Graph the following polar curve: r 3 2sin r 3 2cos 44 P a g e

45 Summary of the Dimpled Limacon: r a bcos For a polar equation to be considered a Dimpled Limacon, the value a must be larger than b. Plug in 0. This will give you the value of r and where you will start the curve moving counterclockwise. There will be no value at the pole. The value of a will tell you where the curve is at on the y-axis 3 (when and ) 2 2 To complete the entire shape 0 2. The value of a b will tell you how far out to the right on the x- axis the curve is if cosine is positive The value of a b will tell you how far out to the left on the x-axis the curve is if cosine is negative The value of a b will tell you how far out on the left of the x- axis the curve is if cosine is positive The value of a b will tell you how far out on the right of the x- axis the curve is if cosine is negative Summary of the Limacon: r a bsin For a polar equation to be considered a Dimpled Limacon, the value a must be larger than b. Plug in 0. This will give you the value of r and where you will start the curve moving counterclockwise. There will be no value at the pole. The value of a will tell you where the curve is at on the x-axis (when 0 and ) To complete the entire shape 0 2. The value of a b will tell you how far up on the y-axis the curve is if sine is positive The value of a b will tell you how far up on the y-axis the curve is if sine is negative The value of a b will tell you how far down on the y-axis the curve is if sine is positive The value of a b will tell you how far up on the y-axis the curve is if sine is negative 45 P a g e

46 Graph the following polar curve: r 2cos 2 r 2cos3 1. What is the value of r when 1. What is the value of r when 0? 0? 2. Divide 360 by the number 2. Divide 360 by the number of petals (first petal) of petals (first petal) 3. Add the number in part 2 3. Add the number in part 2 until you get to 360 (tips of petals) until you get to 360 (tips of petals) Graph the following polar curve: r 3cos 4 r 3cos5 46 P a g e

47 Graph the following polar curve: r 2sin 2 r 2sin3 1. What is the value of r when 1. What is the value of r when 0? 0? divided by number in front divided by number in of (First petal) of (First petal) 3. Divide 360 by the number 3. Divide 360 by the number of petals (distance between tips (distance between tips of petals) of petals) Graph the following polar curve: r 3sin 4 r 3sin5 47 P a g e

48 Summary of the Rose: r acos b For a polar equation to be considered a Rose, the value b must be greater than 1 Plug in 0. This will give you the value of r and where you will start the the curve moving counterclockwise back to the pole. The value of a will tell you how far the furthest point away from the pole is (These occur at the tips of the rose petals) Divide 360 by the number of petals. Keep adding this number until you get back to 360. This will be where the tips of the petals are. If b is an odd number, that is the number of rose petals. If b is an even number, there are 2b number of rose petals. If b is negative the curve will be the same as if b is positive because cosine is an even function. If a is a negative number, the rose petal starts on the left side of the pole If a is a positive number, the rose petal starts on the right side of the pole Summary of the Rose: r asin b For a polar equation to be considered a Rose, the value b must be greater than 1 Plug in 0. This will give you the value of r and where you will start the the curve moving counterclockwise out from the pole. The value of a will tell you how far the furthest point away from the pole is (These occur at the tips of the rose petals) Divide 90 by b. This will be where the end of the first petal is. Divide 360 by the number of petals. Keep adding this number to your first tip until you get back to 360. This will be the angles where the tips of the petals are. If b is an odd number, that is the number of rose petals. If b is an even number, there are 2b number of rose petals. If a and b are both positive or both negative the rose curve will open into the first quadrant. If a or b is negative the rose curve will open into the third quadrant, because sine is an odd function 48 P a g e

49 Lemiscate A) 2 4cos r B) r 2 4sin A) r 2 4cos 2 B) r 2 4sin 2 Summary of Lemniscate: r 2 acos b Figure 8 will be on the x-axis if b = 1 or 2 Summary of Lemniscate: r 2 asin b Figure 8 will be on the y-axis if b = 1 Figure 8 will be in quadrant 1 and 3 if b = 2 49 P a g e

50 Directions: a) Find where each of the curves is when 0. Mark this on the appropriate graph b) Find where each of the curves is when. Mark this on 2 the appropriate graph c) Using your information from parts a and b identify the direction the curve is moving. d) Find when each curve is at the pole. e) Find where the 2 curves intersect. 2. The graphs of the polar curves r 2cos and r 2sin are shown in the figure below for P a g e

51 Directions: a) Find where each of the curves is when 0. Mark this on the appropriate graph b) Find where each of the curves is when. Mark this on 2 the appropriate graph c) Using your information from parts a and b identify the direction the curve is moving. d) Find when each curve is at the pole. e) Find where the 2 curves intersect. 3. The graphs of the polar curves r = 2 and r 3 2cos are shown in the figure below. 51 P a g e

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