Table of Contents Rectangular Coordinates What are Polar Coordinates? Graphing Polar Coordinates Converting Polar to Rectangular Converting Rectangular to Polar Assessment Project
Rectangular CoordinatesWe are accustom to graphing with rectangular coordinates. When we see (3, -4) we know that we must go 3 units right and 4 units down using an xy-plane. Click to here see the motion of the point.
Rectangular CoordinatesFor the point (-4, 6) we would move4 units and 6 units Check your answers
Polar CoordinatesSome things, such as navigation, engineering, and modeling real-world situations can’t easily be measured linearly.We have polar coordinates to describe curves and rotations.
Polar CoordinatesAll polar coordinates begin with a pole (much like the origin). Click here to show the poleAnd a polar axis (think positive x-axis, like where all trig angles start.) Click here to show the polar axis. Polar Axis Pole
Polar CoordinatesPolar Coordinates are written (r, ϴ) where r is the distance from the pole (like a radius) and ϴ is the angle measure from the polar axis.So the point (3, 90º) will look like this. (click here to see animation) 3 units 90º
Polar CoordinatesIf we have the point (2, 135º) click where you think the point will be. Try Again, if your struggling go back to the description. Great Job! (2, 135º)
Polar CoordinatesIf ϴ is negative, travel clockwise, like any other negative angle.Example: (2, -50º) (Click to here see where this will be.) 50º
Polar CoordinatesIf r is negative, this means to move in the opposite direction. So you face the angle where you moved and will travel backwards from there.Example: (-3, 45º) would look like this. (Click here to see motion.) 45º
Polar Coordinates Sorry, Try Again.Click where you would find the point (-3, 60º) Great Job!
Converting Polar toRectangularOften it is useful to be able to go between the two graphing systems. We will use trig to help us convert from polar to rectangular.
Converting Polar toRectangularThink of the point (r, ϴ) anywhere in the polar plane.We can create a right triangle that is x units horizontally and y units vertically. We can call the hypotenuse r because it is the distance from the origin/pole and the angle will be ϴ. (Click here to see the triangle.) (r, ϴ) r y ϴ x
Converting Polar toRectangularUsing trig we know: and (r, ϴ) r y ϴ x
Converting Polar toRectangularSolving both equations for x or y, we get: x = r cos ϴ and y = r sin ϴ We can use both of these equations to convert any point in polar form to rectangular form
Converting Polar toRectangularExample: Convert (4, 135º) from polar form to rectangular form.First: Identify r and ϴ r = 4 and ϴ = 135ºSecond: Find x by plugging r and ϴ x = r cos ϴ into the cosine equation x = 4 cos135º x = 4 (-√2 / 2) x = -2√2 Find y by plugging r and y = r sin ϴThird: ϴ into the sine equation y = 4 sin 135º y = 4 (√2 / 2) y = 2√2 So, we have the point (-2√2, 2√2) in the xy-plane.
Converting Polar toRectangularTry it on your own: Convert (-3, 30º) from polar form to rectangular form. Click on the step number to beginFirst: r= -3 and ϴ= 30 Check step 1 -2.6 ApproximateSecond: x= all fractions to Check step 2 the nearest tenth.Third: y= Check step 3
Converting Rectangular toPolarIn order to convert the other way, from rectangular to polar, we have to use trig and that same right triangle. (x, y) r y ϴ x
Converting Rectangular toPolarSuppose we have the point (x, y). Now we want to find r and ϴ in terms of x and y.In order to find r, we have to find the length of the hypotenuse. (Quietly thank Pythagoras). We knowAnd right triangle trig tells us that (x, y) r y ϴ x
Converting Rectangular to PolarExample: Convert (5, -4) from polar form to rectangular form. Click on the step number to see how this works. x = 5 and y = -4First: Identify x and ySecond: Find r by plugging x and y into the Pythagorean equation Find ϴ by plugging x andThird: y into the tangent equation So, we have the point (√41, 321.3°) in the polar plane.
Converting Rectangular to PolarTry it on your own: Convert (-3, 4) from polar form to rectangular form. Click on the step number to begin.First: x= and y= Check step 1Second: r= Round answers to the nearest tenth. Check step 2Third: ϴ= Check step 3
Converting Rectangular toPolarWhenever your point lies in Quadrant 2 or 3, you must add 180º to your new ϴ.This makes the adjustment that your calculator doesn’t. Your calculator looks for the first ϴ, not necessarily the correct ϴ.
Converting EquationsWhen converting equations from polar form into rectangular form (where we know how to graph it better) we look for the following two equations: x = r cos ϴ and y = r sin ϴAlso be on the lookout for:
Converting Equations Now this looks pretty close to what we Example: want, but there is no r in front of cos ϴ.r *(r = cos ϴ ) So let’s multiply both sides by r. (click here to continue)r2 = r cos ϴ Time to replace what we can, with those equations on the previous slide. (click here to continue)x2 + y2 = x If we complete the square we will get the following equation in standard(x - ½)2 + y2 = ¼ form. (click here to continue) So our equation is a circle with a center at (½, 0) and a radius of ½.
Converting EquationsYour turn: Convert the following equation in polar form into rectangular form. r = 4 sin ϴ 2 2 + = Check your answers
Converting EquationsTry that one more time: Convert the following equation from polar form to rectangular form 4r sin ϴ + 12r cos ϴ = 8 -3x+2 y= Check your answer
AssessmentThe following is a short ten-question quiz.Please click on the letter that best matches the correct answer.
Question 1:In polar coordinates, the origin is called the __________ and the positive x-axis is called the _____________. Check your answers
Sorry, that is incorrect. SeeQuestion 2: Polar Graphing for some help.Which graph below represents the polar coordinate (-2, 330º)? A.) B.) C.) D.) Correct!
Question 3:Which of the following points represents (-3, 215º)? (Click on the appropriate letter.) Sorry, that is Correct! incorrect. A Look B back at graphing. D C Notice that r is negative. Look back at what this means here.
Sorry, that is incorrect.Question 4: Look back at converting.What rectangular coordinates represent the polar coordinates (-2, 30º)? This is still in A.) (-1, -√3) B.) (-√3, -1) polar form. Look back at converting. C.) (-√2, -√2) D.) (2, 210º) Correct!
Sorry, that is incorrect.Question 5: Look back at converting.What is the polar form of the rectangular coordinate (12, 5)? A.) (√119, 67.3º) B.) (13, 67.3º) C.) (√119, 85.2º) D.) (13, 85.2º) Correct!
Sorry, that is incorrect.Question 6: Look back at polar coordinates.What is another polar coordinate that also corresponds to (5, 130º)? A.) (-5, 130º) B.) (-5, 490º) C.) (-5, 310º) D.) (5, -130º) Correct!
Question Sorry, that is incorrect. Look back at converting.7:Find the polar coordinates for the rectangular point (-1.3, -2.1). (Round your answer to the nearest hundredth.)A.) (2.47, 58.24º) B.) (2.47, 238.2º)C.) (-2.47, 238.2º) D.) (2.47, -58.24º) Correct!
Question Sorry, that is incorrect. Look back at converting8: equations.Convert the following equation from polar form to rectangular form. (Select the BEST answer.) r = sin ϴ - cos ϴ A.) 1 = y - x B.) (x + ½) 2 +(y - ½)2 = ½ C.) x 2 + y2 = y - x D.) (x + ¼) 2 +(y - ¼)2 = ½ Correct!
Question Sorry, that is incorrect. Look back at converting equations.9:Which graph represents the equation r = 4? A.) B.) Correct! C.) D.)
Question Sorry, that is incorrect. Look back at converting equations.10:Which graph represents the equation r cos ϴ = 4? A.) B.) Correct! C.) D.)
ProjectRummaging through a friend’s attic, a treasure map was discovered. Lucky day! But at a closer inspection you realize that the map is for Anchorage, Alaska and it is more of a list of directions than a map. It appears that all of the directions are in polar coordinates. But, Anchorage isn’t laid out that way.In 1964, Anchorage was struck by a large earthquake which damaged at great deal of the town. The local government decided it would be best to rebuild from scratch. Roads were designed so that they always crossed at right angles to improve traffic. Anchorage is built like a huge rectangular grid.It appears that the directions begin from your friend’s old family home, which used to be located on the corner of present-day 7th Street and G Street, one block south of the current the Alaska Center of Performing Arts. Your mission is to find the location of the treasure. (Locate the cross-streets on the modern map.) (Search for the corner of 7th and G, Anchorage, AK.)Here are the map directions: Travel (3, 45°) from there, travel (5.83, 210.96°) from there, travel (-3.16, 108.43°) from there (4.47, 386.57°) and finally travel (6.40, - 231.34°).And your follow-up assignment: Paris was designed long before cars and updated traffic constraints. The city is laid out much like a polar grid. With a partner, design a treasure map with a difficult-to-locate solution to challenge another pair. (Include at least four steps and a correct solution on a separate sheet of paper.)
Sources:“Corner of 7th and G Anchorage, AK.” Google Maps. 6 April 2011 <http://maps.google.com/>.Gurewich, Nathan and Ori Gurewich. Teach Yourself Visual Basic 4 in 21 days: Third Edition. Sams Publishing. Indianapolis, IN. 1995.“History of Anchorage, Alaska.” Wikipedia, the free encyclopedia. 5 April 2011 <http://en.wikipedia.org/wiki/History_of_Anchorage, _Alaska>.Sullivan, Michael. Precalculus: Eighth Edition. Pearson: Prentice Hall. Upper Saddle River, NJ. 2008. All graphics drawn and animated by Nicole Slosser using PowerPoint tools.