θ as rectangular coordinates)

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1 Section 11.1 Polar coordinates Learning outcomes After completing this section, you will inshaallah be able to 1. know what are polar coordinates. see the relation between rectangular and polar coordinates 3. learn how to graph polar curves using a. Method I: (from table of values) b. Method II: (by considering r, θ as rectangular coordinates) c. Method III: (by making use of symmetries in above two methods) d. MATLAB 4. know important families of polar curves

2 11.1 What are polar coordinates? Coordinate system: just a way to define a point. In rectangular coordinates a point P is given by coordinates (x,y) which means x (x,y) y start at origin move x units horizontally and y units vertically and we reach point P Another way to define the point P is as ( r, θ ) which means r θ (, r θ ) start at origin move r units along the line making angle θ with positive X-axis Such coordinates are called polar coordinates of point P

3 Standard terminology for polar coordinates Given the polar coordinates ( r, θ ) of a point P Pole (, r θ ) r Polar axis θ θ : Polar angle positive anticlockwise negative clockwise If r > 0 then the point ( r, θ ) lies in the same quadrant as θ. If r < 0 then the point ( r, θ ) lies in the quadrant opposite to that of θ. Example Plot the points Done in class 3,, 3,,,,, Important remark In rectangular coordinate system, each point has unique coordinates but in polar coordinate system a point has infinitely many coordinates Example The coordinates ( r, θ ), ( r, θ n ) + and ( r, θ ( n 1) ) + + all represent the same point. {can you see why?} Example Plot the points Done in class Exercise Do all the coordinates represent the same point? 9 17,,,,, ,, 5,, 5,, 5,

4 Relation between rectangular & polar coordinates See explanation in the class To find rectangular coordinates from polar coordinates we use x = r cosθ (*) y= rsinθ Example Find rectangular coordinates of Done in class 4, 3. To find polar coordinates from rectangular coordinates we use r = x + y 1 y (**) θ = tan x Follow from Eq. (*) Important point to note θ 1 tan y only gives values of < < x But another angle θ + is also possible See example below to understand this point Example Find polar coordinates of ( 1,1 ) and ( 1, 1). Done in class

5 Relation between rectangular & polar coordinates (contd.) The above formulas (*) and (**) can also be used to convert equations from one coordinate system to another. Example Express the following into rectangular coordinates. 1) r = 3 ) r sinθ = 3) r = 3cosθ 4) 6 r = 3cosθ + sinθ Done in class Example Convert x y y + 6 = 0 into polar coordinate system. Done in class

6 Graphing polar curves (Method I) Method I to make graph of r = f( θ ) make a table for values of r, θ plot the points (, r θ ) and join Example Sketch the curve r = cosθ. θ 0 * Table r Note: we have only used values of θ up to because if we take values of θ > we get the same i t * See graph in class Exercise Sketch the curve r = 4sinθ.

7 Graphing polar curves (Method II) Method II to make graph of r = f( θ ) Step I First make the graph of r = f( θ ) in rectangular θ,r - coordinate system (by considering ( r, θ ) as rectangular coordinates) Step II Use the information from Step I to sketch the polar graph of r = f( θ ) Example Sketch the curve r = 1+ cosθ for 0 θ. Step I Graph of r = 1+ cosθ in rectangular coordinates Useful information r 0 for all values of θ r = at 0 θ = 1. As θ increases from 0, r decreases from 1. As θ increases from, r decreases from As θ increases from, r increases from As θ increases from 3, r increases from 1 Step II Draw polar graph of r = 1+ cosθ, using above information. See class notes for the graph

8 Graphing polar curves (Method II Contd.) Example Sketch the curve Step I Graph of r = cosθ in rectangular coordinates r = cosθ for 0 θ. Useful information r = 1 at θ = 0 1. θ changes from 0, r changes from θ changes from, r changes from θ changes from, r changes from θ changes from 3, r changes from θ changes from, r changes from θ changes from 5 3, r changes from θ changes from 3 7, r changes from θ changes from 7, r changes from Means distance from origin increases from 0 to 1 but this portion of curve lies in the quadrant opposite to that of θ. Means distance from origin decreases from 1 to 0 but this portion of curve lies in the quadrant opposite to that of θ. Step II Draw polar graph of r = cosθ, using above information. See class notes for the graph

9 Symmetries of polar curves See explanation in the class to understand the following ideas. A polar curve r = f( θ ) is symmetric about polar axis (or X-axis) if changing θ θ no change in equation e.g. r = cosθ about vertical line θ = (or Y-axis) if changing θ θ no change in equation e.g. r = sinθ about pole (or origin) if changing r r OR θ + θ no change in equation e.g. r = sin θ, r = cosθ Symmetry about pole means curve remains unchanged if we rotate it through 180

10 Graphing polar curves (Method III) Example The graph of using the following table of values θ r = cosθ was sketched in Example by 3 Method III To use symmetry in Method I and Method II r and the graph was Note: r = cosθ is symmetric about polar axis (Why?) and for the values of θ from 0 to we get So we can complete the graph by using symmetry about polar axis and only using the values of θ from 0 to instead of using θ from 0 to.

11 Graphing polar curves (Method III Contd.) Example The graph of r = cosθ was sketched in Example by using θ from 0 to and the graph was Note: r = cosθ is symmetric about pole as well as about polar axis and the line θ = (why?) and for the values of θ from 0 to we get change θ θ + change θ θ change θ θ and check. So we can complete the graph by using symmetries and above part of the curve.

12 Graphing polar curves (Important Exercise) Exercise Consider This consists of two functions r = cosθ for 0 θ. r = cosθ and r = cosθ. a. Find symmetries of r = cosθ. Sketch r = cosθ by using symmetries in Method I Method II b. Find symmetries of r = cosθ. Sketch r = cosθ by using symmetries in Method I Method II c. Do you get same graphs in Part (a) and (b) d. What is the graph of r = cosθ.

13 Before moving on to the next slide See Sections 1 and of the Introductory notes for Matlab beginners. See the handout Plotting graphs in rectangular coordinates using Matlab See the handout Plotting polar curves using Matlab Available in Matlab Help area of WebCT

14 Graphing polar curves r = f( θ ) (using MATLAB) To know when does the curve or r begins to repeat itself Step I Find the domain of θ Find the smallest value of n so that f ( θ + n) = f( θ) Step II Use MATLAB to make the polar graph and take 0 θ n As explained in the handout Plotting polar curves using Matlab Available in Matlab Help area of WebCT

15 Example Use Matlab to plot r = sin 4θ. Step 1 Domain of θ We look at ( n ) sin 4( θ + ) = sin4θ ( n ) sin 4θ + 8 ) = sin 4θ so 0 θ The smallest value of n for which is n = 1. 8n is a multiple of Step Use the following commands to get the graph >> theta=linspace(0,*pi,100); >> r=*sin(4*theta); >> polar(theta,r)

16 Example Use Matlab to plot r = sin5θ. Step 1 Domain of θ We look at ( n ) sin 5( θ + ) = sin5θ ( n ) sin 4θ + 10 ) = sin 4θ so 0 θ The smallest value of n for which is n = 1. 10n is a multiple of Step Use the following commands to get the graph >> theta=linspace(0,*pi,100); >> r=*sin(5*theta); >> polar(theta,r)

17 θ Example Use Matlab to plot r = sin 5. Step 1 Domain of θ Step We look at 8( θ + n) 8θ sin = sin 5 5 8θ 16n 8θ sin + = sin Use the following commands to get the graph >> theta=linspace(0,10*pi,); >> r=sin(8*theta/5); >> polar(theta,r) so 0 θ 10 The smallest value of n for which 16n is a multiple of 5 is n = 5.

18 Example Use Matlab to plot r = + 4cosθ. Step 1 Domain so 0 θ Step >> theta=linspace(0,*pi,100); >> r=+4*cos(theta) >> polar(theta,r) Example Use Matlab to plot r = 4 + cosθ. Step 1 Domain so 0 θ Step >> theta=linspace(0,*pi,100); >> r=4+*cos(theta) >> polar(theta,r)

19 Exercise Use Matlab to plot the following graphs 3θ 1. r = cos. r = 1+ cosθ 3. r = cosθ 4. r = cosθ

20 Important polar graphs Lines 1. θ = α A line that goes through origin and makes an angle α with positive X-axis. rcosθ = a In rectangular coordinates: x = a Hence, a vertical line 3. rsinθ = b In rectangular coordinates: y= b Hence, a horizontal line

21 Circles 1. r = a circle of radius a. r = acosθ circle of radius a with centre at ( a,0) e.g. see graph of r = cosθ in Example r = bsinθ circle of radius b with centre at (0, b ) e.g. see graph of r = 4sinθ in Exercise Exercise Plot a. r = 4 b. r = 4cosθ c. r = 6sinθ

22 11.1 Roses r = acosnθ OR r = asin nθ (n: positive integer) n-even rose with n leaves n-odd rose with n leaves See graph of r = cosθ in Example which had 4 leaves See graph of r = sin5θ in Example which had 5 leaves Observe the following graphs r = cosθ r = sin θ r = cos3θ r = sin3θ n-even Symmetric about X-axis, Y-axis and pole n-odd o cosnθ : Symmetric about X-axis o sin nθ : Symmetric about Y-axis

23 Cardioids r = a± acos θ = a(1 ± cos θ ) r = a± asin θ = a(1 ± sin θ ) heart shaped pass through pole See graph of r = 1+ cosθ in Example Observe the following graphs of cardioids r = a(1 + cos θ ) r = a(1 cos θ ) r = a(1 + sin θ ) r = a(1 sin θ )

24 Limacons r = a± bcosθ r = a± bsinθ ( a< b) have an inner loop pass through pole See graph of r = + 4cosθ in Example Observe the following graphs r = 1+ cosθ r = 1 cosθ r = 1+ sinθ r = 1 sinθ

25 r = a± bcosθ r = a± bsinθ ( a> b) Limacons do not have inner loop do not pass through pole See graph of r = 4 + cosθ in Example Observe the following graphs r = + cosθ r = cosθ r = + sinθ r = sinθ Exercise Plot r = cos θ, r = cos θ, r = + cosθ and compare. Next note the observations explained for Figure in the book.

26 Lemniscates r r = acosθ = asin θ graph with two leaves pass through pole See Exercise Observe the following graphs r = 4cosθ r = 4sinθ

27 Spirals r = aθ End of 11.1 Do Qs: 1-59