6.3 Converting Between Systems

Transcription

1 Chapter 6. The Polar System 6. Converting Between Systems Learning Objectives Convert rectangular coordinates to polar coordinates. Convert equations given in rectangular form to equations in polar form and vice versa. Polar to Rectangular Just as x and y are usually used to designate the rectangular coordinates of a point, r and θ are usually used to designate the polar coordinates of the point. r is the distance of the point to the origin. θ is the angle that the line from the origin to the point makes with the positive x axis. The diagram below shows both polar and Cartesian coordinates applied to a point P. By applying trigonometry, we can obtain equations that will show the relationship between polar coordinates(r,θ) and the rectangular coordinates(x,y) The point P has the polar coordinates(r,θ) and the rectangular coordinates(x,y). Therefore x=r cosθ r = x + y tanθ= y x These equations, also known as coordinate conversion equations, will enable you to convert from polar to rectangular form. Example 1: Given the following polar coordinates, find the corresponding rectangular coordinates of the points: W(4, 00 ),H ( 4, π ) Solution: a) For W(4, 00 ),r=4 and θ=

2 6.. Converting Between Systems x=r cosθ x=4cos( 00 ) y=4sin( 00 ) x=4(.996) x.76 The rectangular coordinates of W are approximately(.76,1.7). b) For H ( 4, π ),r=4 and θ= π y=4(.40) y 1.7 x=r cosθ x=4cos π ( ) 1 x=4 y=4sin π ( ) y=4 x= y= The rectangular coordinates of H are (, ) or approximately(,.46). In addition to writing polar coordinates in rectangular form, the coordinate conversion equations can also be used to write polar equations in rectangular form. Example : Write the polar equation r=4cosθ in rectangular form. Solution: r=4cosθ r = 4r cosθ Multiply both sides by r. x + y = 4x r = x + y and x=r cosθ The equation is now in rectangular form. The r and θ have been replaced. However, the equation, as it appears, does not model any shape with which we are familiar. Therefore, we must continue with the conversion. x 4x+y = 0 x 4x+4+y = 4 Complete the square f or x 4x. (x ) + y = 4 Factor x 4x+4. The rectangular form of the polar equation represents a circle with its centre at (, 0) and a radius of units. 418

3 Chapter 6. The Polar System This is the graph represented by the polar equation r= 4cosθ for 0 θ π or the rectangular form(x ) +y = 4. Example : Write the polar equation r=cscθ in rectangular form. Solution: r=cscθ r cscθ = 1 r cscθ = divide bycscθ r sinθ= sinθ= 1 cscθ y= Rectangular to Polar When converting rectangular coordinates to polar coordinates, we must remember that there are many possible polar coordinates. We will agree that when converting from rectangular coordinates to polar coordinates, one set of polar coordinates will be sufficient for each set of rectangular coordinates. Most graphing calculators are programmed to complete the conversions and they too provide one set of coordinates for each conversion. The conversion of rectangular coordinates to polar coordinates is done using the Pythagorean Theorem and the Arctangent function. The Arctangent function only calculates angles in the first and fourth quadrants so π radians must be added to the value of θ for all points with rectangular coordinates in the second and third quadrants. 419

4 6.. Converting Between Systems In addition to these formulas, r= x + y is also used in converting rectangular coordinates to polar form. Example 4: Convert the following rectangular coordinates to polar form. P(, 5) and Q( 9, 1) Solution: For P(, 5) x= and y= 5. The point is located in the fourth quadrant and x>0. r= x + y θ=arc tan y ( x r= () +( 5) θ=tan 1 5 ) r= 4 θ 1.0 r 5.8 The polar coordinates of P(, 5) are P(5.8, 1.0). For Q( 9, 1) x= 9 and y= 5. The point is located in the third quadrant and x<0. r= x + y θ=arctan y x + π ( ) 1 r= ( 9) +( 1) θ=tan 1 + π 9 r= 5 θ 4.07 r=15 The polar coordinates of Q( 9, 1) are Q(15,4.07) Converting Equations To write a rectangular equation in polar form, the conversion equations of x=r cosθ and are used. Example 5: Write the rectangular equation x + y = x in polar form. Solution: Remember r= x + y,r = x + y and x=r cosθ. 40

5 Chapter 6. The Polar System x + y = x r = (r cosθ) r = r cosθ r=cosθ Pythagorean T heorem and x=r cosθ Divide each side by r Example 6: Write the rectangular equation(x ) + y = 4 in polar form. Solution: Remember x=r cosθ and. (x ) + y = 4 (r cosθ ) +(r sinθ) = 4 x=r cosθ and amp;r cos θ 4r cosθ+4+r sin θ=4 expand the terms amp;r cos θ 4r cosθ+r sin θ=0 subtract 4 f rom each side amp;r cos θ+r sin θ=4r cosθ isolate the squared terms amp;r (cos θ+sin θ)=4r cosθ f actor r a common f actor amp;r = 4r cosθ Pythagorean Identity amp;r=4cosθ Divide each side by r If the graph of the polar equation is the same as the graph of the rectangular equation, then the conversion has been determined correctly. (x ) + y = 4 41

6 6.. Converting Between Systems The rectangular equation (x ) + y = 4 represents a circle with center (, 0) and a radius of units. The polar equation r=4cosθ is a circle with center (, 0) and a radius of units. Converting Using the Graphing Calculator You have learned how to convert back and forth between polar coordinates and rectangular coordinates by using the various formulae presented in this lesson. The TI graphing calculator allows you to use the angle function to convert coordinates quickly from one form to the other. The calculator will provide you with only one pair of polar coordinates for each pair of rectangular coordinates. Example 7: Express the rectangular coordinates of A(,7) as polar coordinates. Polar coordinates are expressed in the form (r,θ). An angle can be measured in either degrees or radians, and the calculator will express the result in the form selected in the MODE menu of the calculator. Press MODE and cursor down to Radian Degree. Highlight Degree. Press nd mode to return to home screen. To access the angle menu of the calculator press nd APPS and this screen will appear: Cursor down to 5 and press ENTER. The following screen will appear 4

7 Chapter 6. The Polar System. Press -, 7) ENTER and the value of r will appear. Access the angle menu again by pressing nd APPS. When the angle menu screen appears, cursor down to 6 and pres ENTER or press 6 on the calculator. The screen will appear. Press -, 7) ENTER and the value of θ will appear. This procedure can be repeated to determine the rectangular coordinates in radians. Before starting, press MODE and cursor down to Radian Degree and highlight Radian. Example 8: Express the polar coordinates of(00,70 ) in rectangular form. The angle θ is given in degrees so the mode menu of the calculator should also be set in degree. Therefore, press MODE and cursor down to Radian Degree and highlight degree. Press nd mode to return to home screen. To access the angle menu of the calculator press nd APPS and this screen will appear: Cursor down to 7 and press ENTER or press 7 on the calculator. The following screen will screen will appear: Press 00, 70) and the value of x will appear Access the angle menu again by pressing nd pres ENTER or press 8 on the calculator. The screen APPS. When the angle menu screen appears, cursor down to 8 and 4

8 6.. Converting Between Systems will appear. Press 00,70) ENTER and the value of y will appear. Points to Consider When we convert coordinates from polar form to rectangular form, the process is very straightforward. However, when converting a coordinate from rectangular form to polar form there are some choices to make. For example the point 0,1 could translate to(1,π) or to (1, 4π), and so on. Are there any advantages to using polar coordinates instead of rectangular coordinates? List any situations in which this is the case. What types of curves are easier to draw with polar coordinates? List situations in which rectangular coordinates are preferable. Review Questions 1. For the following polar coordinates that are shown on the graph, determine the rectangular coordinates for each point. 44. Write the following polar equations in rectangular form. a. r=6cosθ b. r sinθ= c. r=sinθ d. r sin θ=cosθ. Write the following rectangular points in polar form. a. A(, 5) using radians b. B(5, 4) using radians c. C(1, 9) using degrees

9 Chapter 6. The Polar System d. D( 1, 5) using degrees 4. Write the rectangular equations in polar form. a. (x 4) +(y ) = 5 b. x y=1 c. x + y 4x+y=0 d. x = 4y Review Answers 1. For A,r= 4 and θ= 5π 4 x=r cosθ For B,r= and θ=15 x= 4cos 5π ( 4 ) x= 4 y= 4sin 5π ( 4 ) y= 4 x= y= x=r cosθ x= cos15 y= sin15 x= y= x= y= For C, r=5 and θ= ( ) π x=r cosθ 1. x=5cos π ( x=5 1 ) y=5sin π ( ) y=5 x=.5 y= 5 r=6cosθ r = 6r cosθ x + y = 6x x 6x+y = 0 x 6x+9+y = 9 (x ) + y = 9 45

10 6.. Converting Between Systems 4. r sinθ= y= r=sinθ r = r sinθ x + y = y y y= x y y+1= x + 1 (y 1) = x + 1 x +(y 1) = 1 r sin θ=cosθ r sin θ=r cosθ y = x. a. For A(,5)x= and y=5. The point is located in the second quadrant and x<0. r= ( ) +(5) = 9 5.9, θ=arc tan 5 + π=1.95. The polar coordinates for the rectangular coordinates A(, 5) are A(5.9, 1.95) b. For B(5, 4)x = 5 and y= 4. The point is located in the fourth quadrant and x>0. r= (5) +( 4) = ( ) , θ=tan The polar coordinates for the rectangular coordinates B(5, 4) are A(6.40, 0.67) c. C(1, 9) is located in the first quadrant. r= = , θ=tan d. D( 1, 5) is located in the third quadrant and x<0. r= ( 1) +( 5) = 169=1, θ=tan (x 4) +(y ) = 5 x 8x+16+y 6y+9=5 x 8x+y 6y+5=5 x 8x+y 6y=0 x + y 8x 6y=0 From graphing r 8cosθ 6sinθ=0, we see that the r 8(r cosθ) 6(r sinθ)=0 additional solutions are 0 and 8. r 8r cosθ 6r sinθ=0 r(r 8cosθ 6sinθ)=0 r=0 or r 8cosθ 6sinθ=0 r=0 or r=8cosθ+6sinθ 46

11 Chapter 6. The Polar System.. x y=1 r cosθ r sinθ=1 r(cosθ sinθ)=1 1 r= cosθ sinθ x + y 4x+y=0 r cos θ+r sin θ 4r cosθ+r sinθ=0 r (sin θ+cos θ) 4r cosθ+r sinθ=0 r(r 4cosθ+sinθ)=0 r=0 or r 4cosθ+sinθ=0 r=0 or r=4cosθ sinθ 4. x = 4y (r cosθ) = 4(r sinθ) r cos θ=4r sin θ 4r sin θ r cos θ = 1 4tan θsecθ = 1 r 4tan θsecθ=r 47