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