Introduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1)

Transcription

1 POLAR EQUATIONS AND GRAPHS GEOMETRY INU4/54 (MATHS ) Dr Adrin Jnnett MIMA CMth FRAS Polr equtions nd grphs / 6 Adrin Jnnett

2 Objectives The purpose of this presenttion is to cover the following topics: Polr coordinte sstem Conversion between polr nd Crtesin coordintes Sketching polr functions Conversion between Polr equtions nd Crtesin equtions Stndrd polr curves Polr equtions nd grphs / 6 Adrin Jnnett

3 Introduction Polr coordintes re used to uniquel specif points in plne. We re lred used to Crtesin coordintes to specif our points on grphs but there re other ws of doing it. In the polr coordinte sstem ech point is specified b distnce r from the origin nd n ngleθ mesured in n nticlockwise sense from the positive -is. Thus we cn specif the point P in the plne s P(, ) or P(r;θ) using Crtesin or polr coordintes respectivel. Note the convention of using semi-colon (;) to seprte the polr coordintes. Polr equtions nd grphs / 6 Adrin Jnnett

4 Coordinte sstem trnsformtions P r θ The following equtions will trnsform polr coordintes to Crtesin coordintes: = r cosθ nd = r sinθ () To trnsform Crtesin coordintes to polr coordintes, we use Pthgors s theorem: r= + nd θ= tn () where the ngleθ must be given in the correct qudrnt. To uniquel specif ech point on the plne, we must restrict the ngle to the intervl θ< 6 (or θ< π). This is identicl to the method used for vector mgnitude nd direction. Polr equtions nd grphs 4/ 6 Adrin Jnnett

5 Crtesin to polr conversion Give the point P(, 4) in polr coordintes. Drw picture it s visul check tht ou re doing everthing ok. (, 4) 4 r The = nd = 4 vlues give: θ r =( ) + 4 = 5 r= 5 The principl vlue isθ= tn 4 = 5.. The picture shows the point is in the second qudrnt, so we dd 8 to getθ= 6.9. The polr coordintes re(5; 6.9 ). Polr equtions nd grphs 5/ 6 Adrin Jnnett

6 Polr to Crtesin conversion Give the point Q(4; ) in Crtesin coordintes. We re given r= 4 ndθ=. The coordintes re found using =r cosθ nd = r sinθ : = 4 cos = 4 = = 4 sin = 4 = The Crtesin coordintes re Q(, ). Polr equtions nd grphs 6/ 6 Adrin Jnnett

7 Crtesin grph pper 4 4 This is convenient for plotting Crtesin(, ) points nd curves. Polr equtions nd grphs 7/ 6 Adrin Jnnett

8 Polr grph pper This is convenient for plotting polr(r;θ) points nd curves. Polr equtions nd grphs 8/ 6 Adrin Jnnett

9 Plotting polr coordintes 9 6 Instructions for plotting the grph of r= f(θ). 5 Choose vlue forθ nd clculte r using r= f(θ). (Mke tble of vlues). 8 Remember thtθ is tken to be nti-clockwise from the positive horizontl direction. 4 7 We plot the point(r;θ) s follows: Mesure distnce r from the origin in the direction ofθ. Negtive vlues of r re mesured w from the direction ofθ. Polr equtions nd grphs 9/ 6 Adrin Jnnett

10 Plotting polr curve Plot the curve r= cosθ, θ 8 Let s mke simple tble of vlues to help with the plotting. θ r Let s plot the vlues. 6 Remember: negtive r goes w from the ngle! 5 When we re done with plotting points, we drw smooth curve through them. 8 The curve is circle; but tht might not be obvious becuse we plotted onl few points. We ll prove it lter. 4 7 Polr equtions nd grphs / 6 Adrin Jnnett

11 Plotting polr curve Plot the curve r= +cosθ, θ 6 Let s mke simple tble of vlues to help with the plotting. θ r Let s plot the vlues. 6 5 When we re done with plotting points, we drw smooth curve through them. 8 This hert-shped curve is known s crdioid. 4 7 Polr equtions nd grphs / 6 Adrin Jnnett

12 Plotting polr curve Plot the curve r=, θ 6 We don t need tble of vlues for this. There is no θ in the polr eqution. Tht mens r= for ever vlue ofθ! For emple, the points(; ),(; 45 ),(; 9 )... etc., re ll on the curve The eqution r= represents circle (of rdius, centred t the origin). Polr equtions nd grphs / 6 Adrin Jnnett

13 Plotting polr curve Plot the curve r= sinθ, θ 6 Begin, s usul, with tble of vlues to help with the plotting. θ r In this cse we probbl need more points to see the shpe of the curve clerl. Let s plot more points in between those given in the tble (e.g..5, 67.5 ). When we re done, we ll drw smooth curve through the points. In this emple, plotting points to sketch the curve ws not efficient. We ll see better method lter Polr equtions nd grphs / 6 Adrin Jnnett

14 Stndrd Polr Curves We ll finish this section b showing some well known polr curves. O O r= sinθ r= cosθ Polr equtions nd grphs 4/ 6 Adrin Jnnett

15 These curves re clled lemnisctes. r= sin θ r= cos θ Polr equtions nd grphs 5/ 6 Adrin Jnnett

16 These curves re clled polr roses r= sin θ r= cos θ Polr equtions nd grphs 6/ 6 Adrin Jnnett

17 Crdioid (hert-shped) curve Fermt s Spirl O O r= (+cosθ) r = θ Polr equtions nd grphs 7/ 6 Adrin Jnnett

18 This lst one is Archimedes Spirl. π π π π π π 4π r= θ Polr equtions nd grphs 8/ 6 Adrin Jnnett

19 Crtesin equtions nd polr equtions Erlier we sw how to convert coordintes of points between Crtesin nd polr form. We need to lern to do the sme for Crtesin equtions nd polr equtions. The sme trnsformtion equtions ppl but these slightl different rerrngements re often more useful. = r cosθ cosθ= r = r sinθ sinθ= r r = + nd tnθ= Tr to memorise these trnsformtions. We will occsionll need to mke use of trig identities prticulrl the double ngle formule: sinθ sinθ cosθ cos θ cos θ sin θ Polr equtions nd grphs 9/ 6 Adrin Jnnett

20 Polr eqution to Crtesin form Epress r = cos θ s Crtesin eqution. Refer to the previous slide to see cosθ= r : r = r r = Also, r = + so: + = This is Crtesin eqution the reltionship is given in terms of nd. Rerrnge to show this circle! + = ( ) + = 9 4 This circle is centred t(, ) nd hs rdius r=. Compre this to the sketch ou mde erlier! Polr equtions nd grphs / 6 Adrin Jnnett

21 Polr eqution to Crtesin form Epress r = sinθ s Crtesin eqution. First, we need to epress this in terms of sinθ nd cosθ : r = (sinθ cosθ)=4sinθ cosθ We know cosθ= r nd sinθ= r : Also, r = + so: r = 4 = r r r r 4 = 4 ( + ) = 4 This is Crtesin eqution the reltionship is given in terms of nd. Polr equtions nd grphs / 6 Adrin Jnnett

22 Polr eqution to Crtesin form Epress the polr eqution r= tnθ s Crtesin eqution. We know tnθ= : r= Also, r= + so: + = This is Crtesin eqution. But perhps this is better w to write the function: + = or ( + )=4 Polr equtions nd grphs / 6 Adrin Jnnett

23 Crtesin eqution to polr eqution Epress the Crtesin eqution( + ) = s polr eqution. Given tht r = + nd =r cosθ so( + ) = becomes: (r ) = (r cosθ) r 4 r = r cos θ = cos θ Tking squre root would introduce±into the eqution so we ll just leve it in this form. Polr equtions nd grphs / 6 Adrin Jnnett

24 Crtesin eqution to polr form The Crtesin eqution of n ellipse is given b + b = Write this s polr eqution in the form r = f(θ). Given tht = r cosθ nd = r sinθ then r cos θ + r sin θ = b cos r θ + sin θ = b r b cos θ+ sin θ b = This cn be rerrnged to get: r b = b cos θ+ sin θ Tke squre-root:: r= b b cos θ+ sin θ Polr equtions nd grphs 4/ 6 Adrin Jnnett

25 Polr equtions: ngle mesurement So fr, we hve introduced the bsic concepts of polr equtions using ngles mesured in degrees. Polr ngles be prepred to use rdins! Mthemticins cn nlse polr curves with clculus or solve polr equtions with numericl methods. In such cses it is necessr to use rdins for ngle mesurements. Polr equtions nd grphs 5/ 6 Adrin Jnnett

26 Test ourself If ou ve red nd understood the emples in these notes, ou should be ble to nswer the following questions. Write the coordintes( 8, ) in the polr form. Epress(; 7π 6 ) in s Crtesin coordintes. Convert the polr eqution r = 8cosθ to Crtesin form. 4 Write the eqution( + )( ) = in polr form. 5 Plot the grph of r= sin θ, θ 6 9 (8; 8 ) or(8;π). 5 6 ( 6, 6) 5 ( + ) = 8( ) (hint: use cosθ cos θ sin θ ) 8 4 r= secθ+ 4 7 Polr equtions nd grphs 6/ 6 Adrin Jnnett