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Concept Category 15 Polar Equations & Graphs. LT 6B: I can represent complex numbers on the complex plane in rectangular form and polar form. I can explain why the rectangular and polar forms of a given complex number represent the same number. I can graph polar equations using technology.
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Concept Category 15 Polar Equations & Graphs LT 6B: I can represent complex numbers on the complex plane in rectangular form and polar form. I can explain why the rectangular and polar forms of a given complex number represent the same number. I can graph polar equations using technology
I. Polar Coordinate System • Definition A system of coordinates in which the location of a point is determined by its distance from a fixed point at the center of the coordinate space (called the pole) and by the measurement of the angle formed by a fixed line (the polar axis, corresponding to the x -axis in Cartesian coordinates) and a line from the pole through the given point. The polar coordinates of a point are given as ( r, θ), where r is the distance of the point from the pole, and θis the measure of the angle.
B. Visual Polar Angle: The angle formed by the polar axis and the radius vector in a polar coordinate system.
C. Process Coordinate: (r,Θ) Equation: r=cosΘ
Relationship Between Polar & Rectangular Coordinates • Polar to Rectangular Coordinates x = rcosθ and y = rsinθ • Rectangular to Polar r2 = x2 + y2 and tan = y/x (x≠0)
C. Process Sketch the graph of r=cosΘ in the Polar Coordinate System
https://youtu.be/3tNVOhtvPEw E. Application How could we use a logarithmic spiral?
Automatic Lawnmower! You have a perfectly circular lawn in your yard. You would like to devise a method to have your lawn mower automatically mow that section of your yard. You need to give your software a polar equation to map out the position of the lawnmower over time. The lawn is 125 square meters and your lawn mower has a 320 mm cut width. Create an equation to model the best path of the lawn mower as it mows your grass. Use technology to graph and check your model. How many rotations will the lawn mower need to make in your model?
Goal Problems SAT 2, Level 2 Prep Book: Page 69
Practice Recall & Reproduction: Pg. 597 #1, #3, #13 Routine: Pg. 598 #23, 25, 29, 35, 39, 41 Non-Routine: Pg. 617 #23 (see example 3, page 612) Pg. 617 #17 (see example 4, page 616)
Parametric Equations LT 6B Fundamental Skill I can analyze whether the parametric (parameter is time) or rectangular system is the appropriate choice to model a given situation. I can graph and make sense of parametric equations. I can model real world situations with parametric equations.
Diagram what is happening What is the difference? Why is one more challenging than the other?
Problem: Find Mathematical Model How can we graph this scenario? Why care?
I. Parametric • Definition: Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters.”
B. Visual X(t)= 4t2 Y(t)= 3t Set up a table:
C. Process Polar Connection? What do you notice? Example: Given: x(t)=4t and y(t)= t2 How do you eliminate the parameter?
Eliminating Parameters from Parametric Equations: Given the following parametric equations, can you create one equivalent equation without the parameter?
Goal Problems 1. The graph defined by the parametric equations x = cos2t y = 3 sint -1 is : A) a circle B) a hyperbola C) a vertical line D) part of a parabola E) an ellipse 2. A line has parametric equations x=5+t y=7+t where t is the parameter. The slope of the line is: A) D) B) 1 C) E) 7
D. Purpose: applications that involve time as a function of two other variables • Nolan Ryan throws a baseball with an initial speed of 145 ft. per second at an angle of 20° to the horizontal. The ball leaves Nolan Ryan’s hand at a height of 5 ft. • Create an equation or a set of equations that describe the position of the ball as a function of time. • How long is the ball in the air? (Assume the ball hits the ground without being caught) • How far horizontally would the ball travel in the situation described in (b)? • When is the ball at its maximum height? Determine the maximum height of the ball. • Graph the equations to check your answers. Sketch the graph and show the window • Consider the situation present in a game. Nolan Ryan would like the ball to land in the catcher’s mitt (18 ft. from him on the mound) within the strike zone. Assume the strike zone is between 1.5 ft above the ground (knee height) and 3.75 ft above ground (chest height). If he maintains his 145 ft. per second velocity on each pitch, what angle would ensure his pitch hits the very bottom of the strike zone?
Solution • a) • 3.197 seconds • 435.6 ft. • 43.44 ft at 1.55 seconds • f) Angle: -9.86° or 350.12°
Active Practice Concept Category #16 (6.3): Parametric Equations & Graphs (Honors)
