Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal and vetical aes. y In ealie chaptes, we often found the Catesian coodinates of a point on a cicle at a given angle fom the positive hoizontal ais. Sometimes that angle, along with the point s distance fom the oigin, povides a moe useful way of descibing the point s location than conventional Catesian coodinates. Pola Coodinates Pola coodinates of a point consist of an odeed pai, (, θ, whee is the distance fom the point to the oigin, and θ is the angle measued in standad position. Notice that if we wee to gid the plane fo pola coodinates, it would look like the gaph to the ight, with cicles at incemental adii, and ays dawn at incemental angles. Eample 1 Plot the pola point 5 3,. 6 This point will be a distance of 3 fom the oigin, at an angle of 5. Plotting this, 6 Eample Plot the pola point,. 4 Typically we use positive values, but occasionally we un into cases whee is negative. On a egula numbe line, we measue positive values to the ight and negative values to the left. We will plot this point similaly. To stat we otate to an angle of 4. Moving this diection, into the fist quadant, would be positive values. Fo negative values, we move the opposite diection, into the thid quadant. Plotting this:
468 Chapte 8 Note the esulting point is the same as the pola point 5, 4. Ty it Now 1. Plot the following points given in pola coodinates and label them. 3 a. A = 3, b. B =, c. C = 4, 6 4 Conveting Points To convet between pola coodinates and Catesian coodinates, we ecall the elationships we developed back in Chapte 5. Conveting Between Pola and Catesian Coodinates To convet between pola (, θ and Catesian (, y coodinates, we use the elationships cos(θ = = cos(θ y sin(θ = y = sin(θ y tan(θ = + y = θ (, y y Fom these elationship and ou knowledge of the unit cicle, if = 1 and θ =, the 3 pola coodinates would be (, θ = 1,, and the coesponding Catesian 1 3 coodinates ( y, =,. Remembeing you unit cicle values will come in vey handy as you convet between Catesian and pola coodinates.
Section 8. Pola Coodinates 469 Eample 3 Find the Catesian coodinates of a point with pola coodinates (, θ = 5,. To find the and y coodinates of the point, 1 5 = cos( θ = 5cos = 5 = 3 5 3 y = sin( θ = 5sin = 5 = The Catesian coodinates ae 5 5,. 3 Eample 4 Find the pola coodinates of the point with Catesian coodinates ( 3, 4. We begin by finding the distance using the Pythagoean elationship ( 3 + ( 4 = 9 + 16 = = 5 = 5 + y = Now that we know the adius, we can find the angle using any of the thee tig elationships. Keep in mind that any of the elationships will poduce two solutions on the cicle, and we need to conside the quadant to detemine which solution to accept. Using the cosine, fo eample: 3 cos( θ = = 5 3 cos 1 θ =.14 By symmety, thee is a second possibility at 5 θ =.14 = 4.069 Since the point (-3, -4 is located in the 3 d quadant, we can detemine that the second angle is the one we need. The pola coodinates of this point ae (, θ = (5,4.069. Ty it Now. Convet the following. a. Convet pola coodinates (, θ (, = to ( y., b. Convet Catesian coodinates ( y=, (0, 4 to (, θ.
470 Chapte 8 Pola Equations Just as a Catesian equation like y = descibes a elationship between and y values on a Catesian gid, a pola equation can be witten descibing a elationship between and θ values on the pola gid. Eample 5 Sketch a gaph of the pola equation = θ. The equation = θ descibes all the points fo which the adius is equal to the angle. To visualize this elationship, we can ceate a table of values. θ 0 /4 / 3/4 5/4 3/ 7/4 0 /4 / 3/4 5/4 3/ 7/4 We can plot these points on the plane, and then sketch a cuve that fits the points. The esulting gaph is a spial. Notice that the esulting gaph cannot be the esult of a function of the fom y = f(, as it does not pass the vetical line test, even though it esulted fom a function giving in tems of θ. Although it is nice to see pola equations on pola gids, it is moe common fo pola gaphs to be gaphed on the Catesian coodinate system, and so, the emainde of the pola equations will be gaphed accodingly. The spial gaph above on a Catesian gid is shown hee. Eample 6 Sketch a gaph of the pola equation = 3. Recall that when a vaiable does not show up in the equation, it is saying that it does not matte what value that vaiable has; the output fo the equation will emain the same. Fo eample, the Catesian equation y = 3 descibes all the points whee y = 3, no matte what the values ae, poducing a hoizontal line. Likewise, this pola equation is descibing all the points at a distance of 3 fom the oigin, no matte what the angle is, poducing the gaph of a cicle.
Section 8. Pola Coodinates 471 The nomal settings on gaphing calculatos and softwae gaph on the Catesian coodinate system with y being a function of, whee the gaphing utility asks fo f(, o simply y =. To gaph pola equations, you may need to change the mode of you calculato to Pola. You will know you have been successful in changing the mode if you now have as a function of θ, whee the gaphing utility asks fo (θ, o simply =. Eample 7 Sketch a gaph of the pola equation = 4cos( θ, and find an inteval on which it completes one cycle. While we could again ceate a table, plot the coesponding points, and connect the dots, we can also tun to technology to diectly gaph it. Using technology, we poduce the gaph shown hee, a cicle passing though the oigin. Since this gaph appeas to close a loop and epeat itself, we might ask what inteval of θ values yields the entie gaph. At θ = 0, = 4 cos(0 = 4. We would then conside the net θ value when will be 4, which would mean we ae back whee we stated. Solving, 4 = 4cos( θ cos( θ = 1 θ = 0 o θ = This shows us at 0 adians we ae at the point (0, 4, and again at adians we ae at the point (0, 4 having finished one complete evolution. The inteval 0 θ < yields one complete iteation of the cicle. Ty it Now 3. Sketch a gaph of the pola equation = 3sin( θ, and find an inteval on which it completes one cycle. The last few eamples have all been cicles. Net we will conside two othe named pola equations, limaçons and oses. Eample 8 Sketch a gaph of the pola equation = 4 sin( θ +. What inteval of θ values coesponds to the inne loop? This type of gaph is called a limaçon.
47 Chapte 8 Using technology, we can daw a gaph. The inne loop begins and ends at the oigin, whee = 0. We can solve fo the θ values fo which = 0. 0 = 4sin( θ + = 4sin( θ 1 sin( θ = 7 11 θ = o θ = 6 6 This tells us that = 0, o the gaph passes though the oigin, twice on the inteval [0,. 7 11 The inne loop aises fom the inteval θ. This coesponds to whee the 6 6 function = 4 sin( θ + takes on negative values. Eample 9 Sketch a gaph of the pola equation = cos( 3θ. What inteval of θ values descibes one small loop of the gaph? This type of gaph is called a 3 leaf ose. Again we can use technology to poduce a gaph. The inteval [0, yields one cycle of this function. As with the last poblem, we can note that thee is an inteval on which one loop of this gaph begins and ends at the oigin, whee = 0. Solving fo θ, 0 = cos(3θ Substitute u = 3θ 0 = cos( u 3 5 u = o u = o u = Undo the substitution 3 5 3θ = o 3θ = o 3θ = 5 θ = o θ = o θ = 6 6 Thee ae 3 solutions on 0 θ < which coespond to the 3 times the gaph etuns to the oigin, but the fist two solutions we solved fo above ae enough to conclude that one loop coesponds to the inteval θ <. 6
Section 8. Pola Coodinates 473 If we wanted to get an idea of how the compute dew this gaph, conside when θ = 0. = cos(3 θ = cos(0 = 1, so the gaph stats at (1,0. As we found above, at θ = and 6 θ =, the gaph is at the oigin. Looking at the equation, θ y 0 1 1 0 notice that any angle in between and, fo eample at 6 0 0 0 6 θ = 3, poduces a negative : = cos 3 = cos ( = 1. 1 3-1 3 Notice that with a negative value and an angle with teminal side in the fist quadant, the coesponding Catesian point 0 0 0 would be in the thid quadant. Since = cos( 3θ is negative on θ <, this inteval coesponds to the loop of the gaph in the thid quadant. 6 Ty it Now 4. Sketch a gaph of the pola equation = sin( θ. Would you call this function a limaçon o a ose? Conveting Equations While many pola equations cannot be epessed nicely in Catesian fom (and vice vesa, it can be beneficial to convet between the two foms, when possible. To do this we use the same elationships we used to convet points between coodinate systems. Eample 10 Rewite the Catesian equation + y = 6y as a pola equation. We wish to eliminate and y fom the equation and intoduce and θ. Ideally, we would like to wite the equation with isolated, if possible, which epesents as a function of θ. + y = 6y Remembeing + y = we substitute = 6y y = sin(θ and so we substitute again = 6 sin( θ Subtact 6 sin( θ fom both sides 6 sin( θ = 0 Facto ( 6 sin( θ = 0 Use the zeo facto theoem = 6sin( θ o = 0 Since = 0 is only a point, we eject that solution. The solution = 6sin( θ is faily simila to the one we gaphed in Eample 7. In fact, this equation descibes a cicle with bottom at the oigin and top at the point (0, 6.
474 Chapte 8 Eample 11 Rewite the Catesian equation y = 3 + as a pola equation. y = 3 + Use y = sin(θ and = cos(θ sin( θ = 3 cos( θ + Move all tems with to one side sin( θ 3 cos( θ = Facto out ( sin( θ 3cos( θ = Divide = sin( θ 3cos( θ In this case, the pola equation is moe unwieldy than the Catesian equation, but thee ae still times when this equation might be useful. Eample 1 Rewite the pola equation 3 = as a Catesian equation. 1 cos( θ We want to eliminate θ and and intoduce and y. It is usually easiest to stat by cleaing the faction and looking to substitute values that will eliminate θ. 3 = 1 cos( θ Clea the faction ( 1 cos( θ = 3 Use cos(θ = to eliminate θ 1 = 3 Distibute and simplify = 3 Isolate the = 3 + Squae both sides = ( 3 + Use + y = + y = 3 + ( When ou entie equation has been changed fom and θ to and y we can stop unless asked to solve fo y o simplify. In this eample, if desied, the ight side of the equation could be epanded and the equation simplified futhe. Howeve, the equation cannot be witten as a function in Catesian fom. Ty it Now 5. a. Rewite the Catesian equation in pola fom: y =± 3 b. Rewite the pola equation in Catesian fom: = sin( θ
Section 8. Pola Coodinates 475 Eample 13 Rewite the pola equation = sin( θ in Catesian fom. = sin( θ Use the double angle identity fo sine = sin( θ cos( θ y Use cos(θ = and sin(θ = y = Simplify y = Multiply by 3 = y Since 3 ( + y = y This equation could also be witten as 3 / ( y = y + o ( / 3 + y = y = + y, = + y Impotant Topics of This Section Catesian coodinate system Pola coodinate system Plotting points in pola coodinates Conveting coodinates between systems Pola equations: Spials, cicles, limaçons and oses Conveting equations between systems Ty it Now Answes C A B 1.. a. (, θ (, = convets to ( y=, (,0 3 (, θ = 4, o 4, b. ( y, = ( 0, 4 convets to
476 Chapte 8 3. It completes one cycle on the inteval 0 θ <. 4. This is a 4-leaf ose. 5. a. y =± 3 becomes = 3 b. = sin( θ becomes + y = y
Section 8. Pola Coodinates 477 Section 8. Eecises Convet the given pola coodinates to Catesian coodinates. 7 3 7 1. 7,. 6, 3. 4, 6 4 4 4. 4 9, 5. 6, 4 6. 1, 7. 3, 8. ( 5, 9. 3, 6 10., 11. (3, 1. (7,1 Convet the given Catesian coodinates to pola coodinates. 13. (4, 14. (8, 8 15. ( 4, 6 16. ( 5,1 17. (3, 5 18. (6, 5 19. ( 10, 13 0. ( 4, 7 Convet the given Catesian equation to a pola equation. 1. = 3. y = 4 3. y = 4 4. y = 4 5. y 4y + = 6. y 3 + = 7. y = 8. y = 3y Convet the given pola equation to a Catesian equation. 3sin θ = 4cos θ 9. = ( 30. ( 31. = sin 4 ( θ + 7 cos( θ 3. = cos 6 ( θ + 3sin ( θ 33. = sec( θ 34. = 3csc( θ 35. = cos( θ + 36. = 4sec( θ csc( θ
478 Chapte 8 Match each equation with one of the gaphs shown. cos θ sin θ 37. = + ( 38. = + ( 39. = 4 + 3cos( θ 40. 3 4cos( θ = + 41. 5 = 4. = sin ( θ A B C D E F Match each equation with one of the gaphs shown. θ 43. = log ( θ 44. = θcos( θ 45. = cos 46. = sin ( θ cos ( θ 47. = 1+ sin ( 3θ 48. = 1+ sin ( θ A B C D E F
Section 8. Pola Coodinates 479 Sketch a gaph of the pola equation. 3cos θ 4sin 49. = ( 50. = ( θ 51. = 3sin ( θ 5. = 4sin ( 4θ 53. = 5sin ( 3θ 54. = 4sin ( 5θ 55. = 3cos( θ 56. = 4cos( 4θ 57. = + cos( θ 58. = 3+ 3sin ( θ 59. = 1+ 3sin ( θ 60. = + 4cos( θ 1 61. = θ 6. = θ 63. 3 sec( θ = +, a conchoid 64. 1 =, a lituus 1 θ 65. = sin ( θ tan ( θ, a cissoid 66. 1 sin ( θ =, a hippopede 1 This cuve was the inspiation fo the atwok featued on the cove of this book.