6.3 Polar Coordinates


 Darrell Austin
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1 6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard form Then ou label how long r is: EXAMPLE: Plot 4, 4 in the polar coordinate sstem 180 We can convert into degrees b multipling b 4 You will get 15 degrees This is in the second quadrant First draw the angle and then mark off 4 units to represent the radius EXAMPLE: Plot 5, 5 in the polar coordinate sstem We can convert into degrees b multipling b You will get 00 degrees This is in the fourth quadrant First draw the angle and then mark off 5 units to represent the radius EXAMPLE: Plot, 10 in the polar coordinate sstem We alread have it in degrees This is in the second quadrant First draw the angle in standard position Since we have a negative radius, we must plot this differentl Instead of marking along our original angle, we will draw another angle eactl 180 degrees awa from our original angle Then we mark off units
2 EXAMPLE: Plot, in the polar coordinate sstem Section 6 Notes Page 180 We need to convert this into degrees b multipling b You will get 90 degrees First draw the angle in standard position Remember that negative angles are drawn clockwise in standard position Since we have a negative radius instead of marking along our original angle, we will draw another angle eactl 180 degrees awa from our original angle Then we mark off units EXAMPLE: Plot, 4 in the polar coordinate sstem 180 We need to convert this into degrees b multipling b You will get 15 degrees First draw the angle in standard position Remember that negative angles are drawn clockwise in standard position Since we have a negative radius instead of marking along our original angle, we will draw another angle eactl 180 degrees awa from our original angle Then we mark off units Equivalent Angles There are more than one wa to arrive at the same angle For eample in the previous problem, 15 degrees is the same as If we have 10 degrees then this is the same as So for negative angles, just add 60 degrees For positive angles add negative 60 degrees to find the equivalent angle So basicall we can either move clockwise or counterclockwise to arrive at the same angle r, or r, 60 r, or r, 180 EXAMPLE: Given the polar coordinate 5, 00, find an equivalent polar coordinate that has the following characteristics: a) 60 0, r > 0 b) 0 60, r < 0, c) 60 70, r > 0 a) In this problem we are told to work in degrees We want an angle that is negative that will lead us to the same point We are allowed to add or subtract a 60 and that won t change our problem So we can do This is an equivalent angle So our equivalent point is 5, 60 Our r is positive, so we are done tells us that we can add a 180 degrees to our angle and this will change the r to a r Now if we add 180 degrees then we will get an angle more than 60 degrees, so we must subtract 180 degrees: Our equivalent point is: 5, 10 b) Now we want r to be negative The formula above 180
3 Section 6 Notes Page c) We want r to be positive and we need an angle that is more than one revolution, so we just need to add 60 to our angle: Our equivalent point is: 5, 660 EXAMPLE: Given the polar coordinate 4,, find an equivalent polar coordinate that has the following 4 characteristics: a) 0, r > 0 b) 0, r < 0, c) 4, r > 0 a) In this problem we are told to work in radians We want an angle that is negative that will lead us to the same point We are allowed to add or subtract a and that won t change our problem So we can do 5 5 This is an equivalent angle So our equivalent point is 4, Our r is positive, so we 4 are done b) Now we want r to be negative The formula above, r tells us that we can add a pi to our angle and this will change the r to a r Now if we subtract a pi then we will get a negative angle, and our 7 question tells us we must have a positive angle, so we will add pi to our angle: Adding the pi will change our r to a negative so our equivalent point is: 7 4, 4 c) We want r to be positive and we need an angle that is more than one revolution, so we just need to add a to our angle: Our equivalent point is: 4, 4 EXAMPLE: Given the polar coordinate, 10, find an equivalent polar coordinate that has the following characteristics: a) 60 0, r > 0 b) 0 60, r < 0, c) 60 70, r > 0 a) For this problem we are working all in degrees We want an angle that is negative but an r that is positive Our formula sas that we can either add or subtract 180 from our original angle If we add 180 then we won t have a negative angle anmore so we need to subtract: Now we have the equivalent point:, 00 b) We alread have a negative r Now we need a positive angle We don t want to change so we need to add 60 to our original angle: So our equivalent point is:, 40 c) We need r to be positive and our angle needs to be We need to add 180 to change the r into a negative r We are not done et because this is not between 60 degrees and 70 degrees We can add 60 degrees and this won t change our r:
4 Conversion formulas from polar to rectangular coordinates Section 6 Notes Page 4 r cos r sin r EXAMPLE: Convert 5, into a rectangular point We can use the above formulas and plug in a 5 for r and a for We will have: 5cos This equals: 1 5 Now we will find : 5 5, EXAMPLE: Convert, sin This equals: into a rectangular point So our rectangular point is We can use the above formulas and plug in a  for r and a for We will have: cos This 4 4 equals: Now we will find : sin This equals: 4 So our rectangular point is, EXAMPLE: Convert, into a rectangular point We can use the above formulas and plug in a  for r and a for We will have: cos This 1 equals: 1 Now we will find : sin This equals: So our rectangular point is 1, r 1 sin into a rectangular equation First let s multipl the both sides of the equation b r This will allow us to put in substitutions for the sine: r r r sin Now we will replace the r with and we can replace the r sin with Then we have: There is nothing more we can do with this
5 r 4 into a rectangular equation Section 6 Notes Page 5 We can square both sides b r to get: r 16 So 16, which is a circle r into a rectangular equation cos First we can cross multipl to get: r r cos Now replace the r with with an : Not much more to do here and replace the r cos r sin 4cos into a polar equation First let s multipl the both sides of the equation b r This will allow us to put in substitutions for the sine and cosine: r r sin 4r cos Now we will replace the r with and we can replace the r sin with and the r cos with Then we have: 4 On the quiz and test this tpe of question will be given as multiple choice You would notice that our answer above would not appear as one of the choices There is more we can do with this one First set it equal to zero and group all the and terms together: 4 0 Now we can complete the square on both sides: Now we factor and our answer is: ( ) 1 5 This is a circle Conversion formulas from rectangular to polar coordinates r If (, ) is in the first or fourth quadrant, then tan 1 1 If (, ) is in the second or third quadrant, then tan EXAMPLE: Convert, into a polar coordinate Epress our angle in radians We can use the above formulas and plug in a  for and a for This will give us r: ( ) () r will give us r If we plot, we will end up in the second quadrant So we will use 1 tan, so tan 1 This equals: So our polar coordinates are, 4 This
6 Section 6 Notes Page 6 EXAMPLE: Convert, into a polar coordinate Epress our angle in radians We can use the above formulas and plug in a  for and a for This will give us r: This will give us 4 1, we will end up in the 1 4 third quadrant So we will use tan, so tan 1 This equals: So 4 our polar coordinates are 4, ( ) r r, so r = 4 If we plot EXAMPLE: Convert 1, into a polar coordinate Epress our angle in radians 1 We can use the above formulas and plug in a for and a for This will give us r: 1 r This will give us 1 r 1, so r = 1 If we plot, we will end up in the 1 fourth quadrant So we will use tan, so tan 1 This equals: tan 1 Therefore, 1 So our polar coordinates are 1, into a polar equation In this equation we will replace with r and we will replace with r cos We get: r r cos We need to set this equal to zero and solve for r We will get: r r cos 0 Now factor out an r: r ( r cos ) 0 Solving for r we get: r 0 and r cos 4 1 into a polar equation In this equation we will replace with r cos and we will replace with r sin We get: 4r cos r sin 1 This equals 4r cos sin 1 Can t do much more with this into a polar equation In this equation we will replace with r sin and we will replace with r cos You will get: r sin r cos On the quiz and test this tpe of question will be given as multiple choice You would notice that our answer above would not appear as one of our choices That means we need to simplif this
7 Section 6 Notes Page 7 further First set this equal to zero: sin cos r r 0 Now factor our an r: r ( r sin cos ) 0 Now set both factors equal to zero Y ou will get r = 0 and r sin cos 0 We need to solve the second cos cos 1 equation for r You will get: r This can be written as: r Then our final answer sin sin sin is r cot csc
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