Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Polar Coordinate System by CHED on June 15, 2017 lesson duration of 2 minutes under Precalculus generated on June 15, 2017 at 11:47 am Tags: Trigonometry CHED.GOV.PH K-12 Teacher's Resource Community Generated: Jun 15,2017 07:47 PM Polar Coordinate System ( 2 mins ) Written By: CHED on July 5, 2016 Subjects: Precalculus Tags: Trigonometry Resources N/A N/A Content Standard Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate system Performance Standard Formulate and solve accurately situational problems involving circular functions Apply appropriate trigonometric identities in solving situational problems Formulate and solve accurately situational problems involving appropriate trigonometric functions Formulate and solve accurately situational problems involving the polar coordinate system Learning Competencies Solve situational problems involving polar coordinate system Introduction 1 mins Two-dimensional coordinate systems are used describe a point a plane. We previously used Cartesian Two-dimensional coordinate systems are used to to describe a point in in a plane. We previously used thethe Cartesian or or rectangularcoordinate coordinatesystem systemtotolocate locatea apoint pointin inthe theplane. plane.That Thatpoint pointis isdenoted denotedbyby( (x x, y), where x is is the the signed signed rectangular distance of the point from the y-axis, and y is the signed distance of the point from the x-axis. We sketched the graphs of equations (lines, circles, parabolas, ellipses, hyperbolas) functions (polynomial, rational, exponential, of equations (lines, circles, parabolas, ellipses, andand hyperbolas) andand functions (polynomial, rational, exponential, logarithmic, logarithmic, trigonometric, trigonometric, and and inverse inverse trigonometric) trigonometric) in in the the Cartesian Cartesian coordinate coordinate plane. plane. However, However, itit is is often often convenient convenient to locate a point based on its distance from a fixed point and its angle with respect to a fixed ray. Not all equations can begraphed graphedeasily easilyusing usingCartesian Cartesiancoordinates. coordinates.InInthis thislesson, lesson,we wealso alsouse useanother anothercoordinate coordinatesystem, system,which whichcan canbe be be presented in dartboard-like plane as shown below. 1 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Polar Coordinated of a Point We now introduce the polar coordinate system. system. It is composed of a fixed point called the pole (which is the origin in the Cartesian coordinate system) and a fixed ray called the polar axis (which is the nonnegative x-axis). In the polar coordinate system, a point is described by the ordered pair (r, ?), where the radial coordinater coordinater refers to the directed distance of the point from the pole and the angular coordinate ? refers to a directed angle (usually in radians) from the polar axis to the segment joining the point and the pole. Because a point in polar coordinate system is described by an order pair of radial coordinate and angular coordinate, it willbebemore moreconvenient convenienttotogeometrically geometricallypresent presentthe thesystem systeminina a polar plane plane, ,which whichserves servesjust justlike likethe theCartesian Cartesian will plane.In Inthe thepolar polarplane planeshown shownbelow, below,instead insteadof ofrectangular rectangulargrids gridsininthe theCartesian Cartesianplane, plane,we wehave haveconcentric concentriccircles circles plane. common center at the to identify easily distance (radial coordinate) angular withwith common center at the polepole to identify easily the the distance fromfrom the the polepole (radial coordinate) andand angular raysrays emanating from the pole to show the angles from the polar axis (angular coordinate). 2 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Example 3.9.1. 3.9.1. Plot Plot the the following following points points in in one one polar polar plane: plane: A(3, pi/3), B(1, 5pi/6), C(2, 7pi/6), D(4, 19pi/12), E(3, ?pi), Example F(4, ?7pi/6), G(2.5, 17pi/4 ), H(4, 17pi/6), and I(3,?5pi/3). Solution. As As seen seen in in the the last last example, example, unlike unlike in in Cartesian Cartesian plane plane where where aa point point has has aa unique unique Cartesian Cartesian coordinate coordinate representation, representation, point in in polar polar plane plane have have infinitely infinitely many many polar polar coordinate coordinate representations. representations. For For example, example, the the coordinates coordinates (3, (3, 4) 4) in in the the aa point Cartesian to exactly in plane, the plane, this particular hasrectangular no rectangular coordinate Cartesian planeplane referrefer to exactly one one pointpoint in the and and this particular pointpoint has no coordinate representationsother otherthan than(3, (3,4). 4).However, However,the thecoordinates coordinates(3, (3,pi/3) pi/3)ininthe thepolar polarplane planealso alsorefer refertotoexactly exactlyone onepoint, point, representations butthis thispoint pointhas hasother otherpolar polarcoordinate coordinaterepresentations. representations.For Forexample, example,the thepolar polarcoordinates coordinates(3,?5pi/3), (3,?5pi/3),(3, (3,7pi/3), 7pi/3),(3, (3, but 13pi/3), and (3, 19pi/3) all refer to the same point as that of (3, pi/3). 3 / 27 CHED.GOV.PH K-12 Teacher's Resource Community In In polar polar coordinate coordinate system, system, itit is is possible possible for for the the coordinates coordinates (r, (r, ?) ?) to to have have aa negative negative value value of of r. In this case, the point is |r| units from the pole in the opposite direction of the terminal side of ?, as shown in Figure 3.36. Example 3.9.2. Plot the following points in one polar plane: A(?3, 4pi/3), B(?4, 11pi/6), C(?2,?pi), and D(?3.5,?7pi/4). Solution. As described above, a polar point with negative radial coordinate lies on the opposite ray of the terminal side of ?. 4 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Seatwork/Homework 3.9.1 1. Plot the following points in one polar plane: Answer: 2. Give the polar coordinates (r, ?) with indicated properties that represent the same point as the given polar coordinates 5 / 27 CHED.GOV.PH K-12 Teacher's Resource Community From Polar to Rectangular, and Vice Versa Wenow nowhave havetwo twoways waystotodescribe describepoints pointsonona aplane plane– –whether whethertotouse usethe theCartesian Cartesiancoordinates coordinates( (x x, y) or the polar We coordinates (r, ?). We now derive the conversion from one of these coordinate systems to the other. We superimpose the Cartesian and polar planes, as shown in the following diagram. is represented represented by by the the polar polar coordinates coordinates ((rr,, ?). ?). From From Lesson Lesson 3.2 3.2 (in (in particular, particular, the the boxed boxed definition definition Suppose a point P is on page 138), we know that x = r cos ? and y = r sin ?. Example 3.9.3. Convert the polar coordinates (5, pi) and (?3, pi/6) to Cartesian coordinates. Solution. 6 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Teaching Notes One can can also also easily easily convert convert the the polar polar coordinates coordinates (5,pi) (5,pi) to to its its corresponding corresponding rectangular rectangular coordinates coordinates (?5, (?5, 0) 0) by by simply simply One plotting the point. As explained on page 254 (right after Example 3.9.1), we expect that there are infinitely many polar coordinate representations that correspond to just one given rectangular coordinate representation. Although we can actually determine all of them, we only need to know one of them and we can choose r ) 0. Suppose a point P is represented by the rectangular coordinates (x, y). Referring back to Figure 3.37, the equation of the circle is x^2 + y^2 = r^2 => r = square root of (x^2 + y^2). and the the point point is is the the pole. pole. The The pole pole has has coordinates coordinates (0, (0, ?), ?), where where ?? is is any any We now determine ?. If x = y = 0, then r == 00 and real number. If x == 00 and and yy =/= =/= 0, 0, then then we we may may choose choose ?? to to be be either either pi/2 pi/2 or or 3pi/2 3pi/2 (or (or their their equivalents) equivalents) depending depending on on whether whether yy >> 00 or y < 0, respectively. Now, suppose x =/= 0. From the boxed definition again on page 138, we know that tan ? = y/x where ? is an angle in standard position whose terminal side passes through the point (x (x, y). 7 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Example 3.9.4. Convert each Cartesian coordinates to polar coordinates (r, ?), where r > 0. (1) (?4, 0) (2) (4, 4) (3) (?3,?square root of 3) (4) (6,?2) (5) (?3, 6) (6) (?12,?8) Solution. (1) (?4, 0) ?> (4, pi) Teaching Notes Plotting the points on the superimposed Cartesian and polar planes is a quicker approach in converting rectangular coordinates to polar. 8 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Teaching Notes Recall that tan^?1(?2) is in QIV. Teaching Notes We may also use ? = tan^?1 2/3 ? pi. Seatwork/Homework 1. Convert each polar coordinates to Cartesian coordinates. 9 / 27 CHED.GOV.PH K-12 Teacher's Resource Community 2. Convert each Cartesian coordinates to polar coordinates (r, ?), where r > 0. From the preceding session, we learned how to convert polar coordinates of a point to rectangular and vice versa using the following conversion formulas: r2 = x2 + y2, tan 0 = y/x, x = r cos 0, and y = r sin 0. Because a graph is composed of points, we can identify the graphs of some equations in terms of r and 0. As a quick illustration, the polar graph of the equation r = 2 ? 2sin 0 consists of all points (r (r, 0) that satisfy the equation. Some of these points are (2, 0), (1, pi/6), (0, pi/2), (2,pi), and (4, 3pi/ 2). Example 3.9.5. Identify the polar graph of r = 2, and sketch its graph in the polar plane. Solution. Squaring the equation, we get r2 = 4. Because r2 = x2 + y2, we have x2 + y2 == 4, 4, which which is is aa circle circle of of radius radius 22 andwith withcenter centeratatthe theorigin. origin.Therefore, Therefore,the thegraph graphofof r = =2 2is isa acircle circleofofradius radius2 2with withcenter centeratatthe thepole, pole,asasshown shown and below. 10 / 27 CHED.GOV.PH K-12 Teacher's Resource Community In the previous example, instead of using the conversion formula r2 = x2+y2, we may also identify the graph of r = 2 by observingthat thatits itsgraph graphconsists consistsofofpoints points(2, (2, 0)) for for all all 0. .InInother otherwords, words,the thegraph graphconsists consistsofofall allpoints pointswith withradial radial observing distance 2 from the pole as 0 rotates around the polar plane. Therefore, the graph of r = 2 is indeed a circle of radius 2 as shown. Example 3.9.6. Identify and sketch the polar graph of 0 = ?5pi/4. Solution. The graph of 0 = ?5pi/4 consists of all points (r (r, ?5pi/4) for rER. If r > 0, then points (r (r, ?5pi/4) determine a ray from the pole with angle ?5pi/4 from the polar axis. If r = 0, then (0,?5pi/4) is the pole. If r < 0, then the points (r, ?5pi/4) determine a ray in opposite direction to that of r > 0. Therefore, the graph of ? = ?5?4 is a line passing through the pole and with angle ?5?4 with respect to the polar axis, as shown below. Example 3.9.7. Identify (and describe) the graph of the equation r = 4sin 0. Solution. 11 / 27 CHED.GOV.PH K-12 Teacher's Resource Community r = 4sin 0 r2 = 4r 4r sin 0 x2 + y2 = 4y x2 + y2 ? 4y = 0 x2 + (y ? 2)2 = 4 Therefore, the graph of r = 4sin 0 is a circle of radius 2 and with center at (2, pi/2). Example 3.9.8. Sketch the graph of r = 2? 2 sin 0. Solution. We construct a table of values. 12 / 27 CHED.GOV.PH K-12 Teacher's Resource Community This heart-shaped curve is called a cardioid. Example 3.9.9. The sound-pickup capability of a certain brand of microphone is described by the polar equation r = ?4 cos 0, where |r |r| gives the sensitivity of the microphone to a sound coming from an angle 0 (in radians). (1) Identify and sketch the graph of the polar equation. (2) Sound coming from what angle 0E [0, pi] is the microphone most sensitive to? Least sensitive? Solution. (1) r = ?4 cos 0 r2 = ?4 cos 0 x2 + y2 = ?4x ?4x x2 + 4x 4x + y2 = 0 (x + 2)2 + y2 = 4 This is a circle of radius 2 and with center at (2, pi). 13 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (2) We construct a table of values. From the table, the microphone is most sensitive to sounds coming from angles ? = 0 and 0 = pi, and least sensitive to sound coming from an angle 0 = pi/2. Seatwork/Homework 3.9.3 1. Identify (and describe) the graph of each polar equation. (a) 0 = 2pi/3 Answer: Line passing through the pole with angle 2pi/3 with respect to the polar axis (b) r = ?3 Answer: Circle with center at the pole and of radius 3 (c) r = 2sin 0 Answer: Circle of radius 1 and with center at (1, pi/2) (d) r = 3cos ? Answer: Circle of radius 1.5 and with center at (1.5, 0) (e) r = 2+2cos 0 Answer: A cardioid 2. Sketch the graph of each polar equation. 14 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (a) r = ?3 (b) r = ?2 sin 0 (c) r = 2 + 2sin 0 15 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (d) r = 4cos 0 3. The sound-pickup capability of a certain brand of microphone is described by the polar equation r = 1.5(1 + cos ?), where |r |r| gives the sensitivity of the microphone of a sound coming from an angle 0 (in radians). (a) Identify and sketch the graph of the polar equation. Answer: A cardioid 16 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (b) Sound coming from what angle 0E [0, 2pi) is the microphone most sensitive to? Least sensitive? Answer: Most sensitive at 0 = 0; least sensitive at 0 = pi Exercises 1 mins 1. Plot the following points in one polar plane: Answer: 17 / 27 CHED.GOV.PH K-12 Teacher's Resource Community 2. Give the polar coordinates (r, given polar coordinates. ) with indicated properties that represent the same point as the 3. Convert each polar coordinates to Cartesian coordinates. 18 / 27 CHED.GOV.PH K-12 Teacher's Resource Community 4. Convert each Cartesian coordinates to polar coordinates (r, 2 ), where r ) 0 and 0 ? . 5. Identify and sketch the graph of each polar equation. (a) =– /3 19 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Answer: A line passing through the pole and with angle – /3 with respect to the polar axis (b) r = –3 sin Answer: A circle tangent to the x-axis with center at (0,?1.5) 20 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (c) r = cos Answer: A circle tangent to the y-axis with center at (0.5, 0) 21 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (d) r = 2– 2 cos Answer: A cardioid 22 / 27 CHED.GOV.PH K-12 Teacher's Resource Community (e) r = 1+sin Answer: A cardioid 23 / 27 CHED.GOV.PH K-12 Teacher's Resource Community 6. The graph of the polar equation r = 2cos2 is a four-petaled rose. Sketch its graph. Answer: 24 / 27 CHED.GOV.PH K-12 Teacher's Resource Community *7. A comet travels on an elliptical orbit that can be described by the polar equation r = 1.164 / 1 + 0.967 sin with respect to the sun at the pole. Find the closest distance between the sun and the comet Answer: Closest distance occurs when sin = 1, so r = 1.164 / 1.967 ? 0.59 units 25 / 27 CHED.GOV.PH K-12 Teacher's Resource Community *8. Polar equations are also used by scientists and engineers to model motion of satellites orbiting the Earth. One satellite follows the path r = 36210 / 6 – cos , where r is the distance in kilometers between the center of the Earth and the satellite, and angular measurement in radians with respect to a fixed predetermined axis (a) At what value of closest distance? 2 [0, 2 is the ) is the satellite closest to Earth, and what is the 26 / 27 CHED.GOV.PH K-12 Teacher's Resource Community Answer: The satellite is closest to Earth when cos = = –1, and this occurs when . The closest distance is, therefore, r = 36210 6–(–1) 5182.86 kilometers. (b) How far away from Earth can the satellite reach? Answer: The satellite can reach as far as r = 36210 6–1 7242 km away from the Earth 9. The graph of the polar equation 27 / 27 Powered Poweredby byTCPDF TCPDF(www.tcpdf.org) (www.tcpdf.org)