Converting Between Rectangular and Polar Coordinates
• Sometimes we want to change from rectangular coordinates to polar coordinates, and from polar coordinates to rectangular coordinates.
• The drawing below will help us derive some conversion formulas.
sin y
rsin y r
cos x
rcos x r
2 2 2 x y r
tan y
x
• To change to rectangular coordinates, use the formulas …
siny r
cosx r
• To change to polar coordinates, use the formulas …
2 2 2 r x y
tan y
x
0r
Example 1:Change to rectangular coordinates: 4,
4
siny r
cosx r
4sin4
242
2 2
4cos4
242
2 2
2 2,2 2
Example 2:Change to rectangular coordinates: 2
2,3
siny r
cosx r
22sin
3
3
22
3
22cos
3
1
22
1
1, 3
Example 3:Change to polar coordinates: 4,4
2 2 2 r x y
0r
2 24 4 16 16 32
2 32 r 4 2 r
• Note: when we solve an equation by taking the square root of both sides, we usually write a ±, but in this case we always want a non-negative value for r.
Example 3:Change to polar coordinates: 4,4
tan y
x
4
4
1 3
4
• Note: there were two possible values for θ, but we chose the positive value, since the rectangular point is in quadrant II.
Example 3:Change to polar coordinates: 4,4
• The polar point is given by …
34 2,
4
Example 4:Change to polar coordinates: 1, 3
2 2 2 r x y
0r
221 3 1 3 4
2 4 r 2 r
Example 4:Change to polar coordinates:
tan y
x
3
1
3 2
3
• Note: there were two possible values for θ, but we chose the negative value, since the rectangular point is in quadrant III.
1, 3
Example 4:Change to polar coordinates:
• The polar point is given by …
22,
3
1, 3
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