Using Polar Coordinates Graphing and converting polar and rectangular coordinates Butterflies are among the most celebrated of all insects. It’s hard not to notice their beautiful colors and graceful flight. Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system. In many cases, polar coordinates are simpler and easier to use than rectangular coordinates. We are going to look at a You are familiar with new coordinate system plotting with a rectangular called the polar coordinate system. coordinate system. The center of the graph is Angles are measured from called the pole. the positive x axis. Points are represented by a radiusradius and an angle (r, ) To plot the point 5, 4 First find the angle Then move out along the terminal side 5 The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r, ). P = (r, ) O = directed angle Polar Pole (Origin) axis r is the directed distance from O to P. is the directed angle (counterclockwise) from the polar axis to OP. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Graphing Polar Coordinates The grid at the left is a polar grid. The A typical angles of 30o, 45o, 90o, … are shown on the graph along with circles of radius 1, 2, 3, 4, and 5 units. Points in polar form are given as (r, ) where r is the radius to the point and is the angle of the point. On one of your polar graphs, plot the point (3, 90o)? The point on the graph labeled A is correct. A negative angle would be measured clockwise like usual. 3 3, 4 To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side. 2 4, 3 Polar coordinates can also be given with the angle in degrees. 90 (8, 210°) 120 60 135 45 150 30 (6, -120°) 180 0 330 210 (-5, 300°) 315 225 300 240 270 (-3, 540°) Graphing Polar Coordinates 3 Now, try graphing 2 , . 4 A C Did you get point B? Polar points have a new aspect. A radius can be negative! A negative radius means to go in the exact opposite B direction of the angle. To graph (-4, 240o), find 240o and move 4 units in the opposite direction. The opposite direction is always a 180o difference. Point C is at (-4, 240o). This point could also be labeled as (4, 60o). Graphing Polar Coordinates How would you write point A with a negative radius? A C A correct answer would be (-3, 270o) or (-3, -90o). In fact, there are an infinite number of ways to label a single polar point. B Is (3, 450o) the same point? Don’t forget, you can also use radian angles as well as angles in degrees. On your own, find at least 4 different polar coordinates for point B. Plotting Points The point ( r , ) 2, lies two units from the pole on the 3 terminal side of the angle . 3 2 3 2, 3 1 2 3 0 3, 3 4 3 4 3 units from 3 the pole 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 2, . There are many ways to represent the point 3 2, 5 2 3 2, 2, 5 2, 24 2, 3 3 33 2, 3 additional ways 0 to represent the 1 2 3 point 2, 3 (r , ) r , 2 n 3 (r , ) r , (2 n 1) 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Let's plot the following points: 7, 7, 2 2 5 3 7, 7, 2 2 Notice unlike in the rectangular coordinate system, there are many ways to list the same point. Converting from Rectangular to Polar Find the polar form for the rectangular point (4, 3). To find the polar coordinate, we must calculate the radius and angle to the (4, 3) given point. r We can use our knowledge of right 3 triangle trigonometry to find the radius and angle. 4 r2 = 32 + 42 tan = ¾ r2 = 25 = tan-1(¾) r = 5 = 36.87o or 0.64 rad The polar form of the rectangular point (4, 3) is (5, 36.87o) Converting from Rectangular to Polar In general, the rectangular point (x, y) is converted to polar form (r, θ) by: 1. Finding the radius r2 = x2 + y2 (x, y) r 2. Finding the angle y tan = y/x or = tan-1(y/x) Recall that some angles require x the angle to be converted to the appropriate quadrant. Converting from Rectangular to Polar On your own, find polar form for the point (-2, 3). 3 tan (-2, 3) r2 = (-2)2 + 32 2 r2 = 4 + 9 1 3 2 tan r = 13 2 r = 56.31o However, the angle must be in the second quadrant, so we add 180o to the answer and get an angle of 123.70o. The polar form is ( 13 , 123.70o) Converting from Polar to Rectanglar Convert the polar point (4, 30o) to rectangular coordinates. We are given the radius of 4 and angle of 30o. Find the values of x and y. Using trig to find the values of x and y, we know 4 that cos = x/r or x = r cos . Also, sin = y/r or y y = r sin . 30o x r cos y rsin x o o y 4sin30 x 4cos30 y 41 2 3 2 x 4 2 3 2 The point in rectangular form is: 2 3, 2 Converting from Polar to Rectanglar On your own, convert (3, 5π/3) to rectangular coordinates. We are given the radius of 3 and angle of 5π/3 or 300o. Find the values of x and y. o x r cos -60 y rsin o o x 3cos300 y 3sin300 1 3 3 3 x 3 y 3 3 2 2 2 2 3 3 3 The point in rectangular form is: , 2 2 The relationship between rectangular and polar coordinates is as follows. y The point (x, y) lies on a circle of radius r, therefore, r2 = x2 + y2. (x, y) Definitions of (r, ) trigonometric functions r y sin y r cos x Pole r x y tan (Origin) x x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 Coordinate Conversion xr cos yr sin r2 x 2 y 2 (Pythagorean Identity) Example: Convert the point 4, into rectangular coordinates. 3 y sin xrcos 4 cos4 1 2 r 3 2 cos x r 3 y yrsin 4 sin 4 2 3tan 3 2 x xy, 2, 2 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 Example: Convert the point (1,1) into polar coordinates. xy, 1,1 y tan 1 1 x 1 4 r x2 y 2 11 2 2 2 One set of polar coordinates is ( r , ) 2, . 4 Another set is (r , ) 2,5 . 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. (3, 4) Based on the trig you know can you see r how to find r and ? 4 32 42 r 2 3 r = 5 4 tan 3 We'll find in radians 1 4 polar coordinates are: (5, 0.93) tan 0.93 3 Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) x2 y2 r 2 r y r x2 y2 x y tan x y tan1 x Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see 4, how to find x and y? 4 x 4 y cos 4 x 4 4 2 x 4 2 2 2 y sin rectangular coordinates are: 4 4 2 2 2, 2 2 y 4 2 2 2 Let's generalize the conversion from polar to rectangular coordinates. x cos r r, r x r cos y x y sin r y r sin Convert the rectangular coordinate system equation to a polar coordinate system equation. From conversions, how 2 2 r 3 x y 9 was r related to x2 and y2 ? Here each r unit is 1/2 and 2 2 we went out 3 r x y and did all angles. r must be 3 but there is no Before we do the conversion restriction on so consider let's look at the graph. all values. Convert the rectangular coordinate system equation to a polar coordinate system equation. x2 4y What are the polar conversions substitute in for we found for x and y? x and y r cosx rcos2 4r sin r 2 cos2 4r sin We wouldn't recognize what this equation looked like in polar coordinates but looking aty the rrectangularsin equation we'd know it was a parabola. When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular coordinates. r 7 We could square both sides r 2 49 Now use our conversion: r 2 x2 y2 x2 y2 49 We recognize this as a circle with center at (0, 0) and a radius of 7. On polar graph paper it will centered at the origin and out 7 Let's try another: Take the tangent of both sides 3 To graph on a polar plot tan tan 3 we'd go to where 3 and make a line.