Cartesian to Polar worksheet

1. Cartesian point (x, y): (−5.735764, 8.19152)

1. The x-coordinate is the across part of the distance: multiply r by the cosine of θ

   $x$$=10\times \text{cos}125\text{°}$$=\mathrm{-5.735764}$

2. The y-coordinate is the up part: multiply r by the sine of θ

   $y$$=10\times \text{sin}125\text{°}$$=8.19152$

3. Putting x and y together, the Cartesian point is

   $\text{(−5.735764, 8.19152)}$

2. Cartesian point (x, y): (−0.940756, 8.950697)

1. The x-coordinate is the across part of the distance: multiply r by the cosine of θ

   $x$$=9\times \text{cos}96\text{°}$$=\mathrm{-0.940756}$

2. The y-coordinate is the up part: multiply r by the sine of θ

   $y$$=9\times \text{sin}96\text{°}$$=8.950697$

3. Putting x and y together, the Cartesian point is

   $\text{(−0.940756, 8.950697)}$

3. Polar point (r, θ): (5.830952, 120.963757°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-3}\right)}^2+5^2}$$=5.830952$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}5\text{,}\mathrm{-3}\text{)}$$=120.963757\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=120.963757\times \frac{\pi }{180}$$=2.111216$

4. Putting r and θ together, the polar coordinates are

   $\text{(5.830952, 120.963757°)}$

4. Polar point (r, θ): (6.403124, 128.659808°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-4}\right)}^2+5^2}$$=6.403124$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}5\text{,}\mathrm{-4}\text{)}$$=128.659808\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=128.659808\times \frac{\pi }{180}$$=2.245537$

4. Putting r and θ together, the polar coordinates are

   $\text{(6.403124, 128.659808°)}$

5. Polar point (r, θ): (5, 126.869898°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-3}\right)}^2+4^2}$$=5$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}4\text{,}\mathrm{-3}\text{)}$$=126.869898\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=126.869898\times \frac{\pi }{180}$$=2.214297$

4. Putting r and θ together, the polar coordinates are

   $\text{(5, 126.869898°)}$

6. Polar point (r, θ): (5.385165, 111.801409°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-2}\right)}^2+5^2}$$=5.385165$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}5\text{,}\mathrm{-2}\text{)}$$=111.801409\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=111.801409\times \frac{\pi }{180}$$=1.951303$

4. Putting r and θ together, the polar coordinates are

   $\text{(5.385165, 111.801409°)}$

7. Polar point (r, θ): (7.211103, 123.690068°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-4}\right)}^2+6^2}$$=7.211103$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}6\text{,}\mathrm{-4}\text{)}$$=123.690068\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=123.690068\times \frac{\pi }{180}$$=2.158799$

4. Putting r and θ together, the polar coordinates are

   $\text{(7.211103, 123.690068°)}$

8. Polar point (r, θ): (5.656854, 135°)

1. The radius r is the distance from the center. x, y and r form a right triangle, so the Pythagorean theorem gives it

   $r$$=\sqrt{{\left(\mathrm{-4}\right)}^2+4^2}$$=5.656854$

2. Find the angle with atan2, a version of the inverse tangent that checks the signs of x and y to put θ in the right quarter

   $\theta$$=\text{atan2(}4\text{,}\mathrm{-4}\text{)}$$=135\text{°}$

3. To get radians, multiply the degrees by π/180

   $\theta$$=135\times \frac{\pi }{180}$$=2.356194$

4. Putting r and θ together, the polar coordinates are

   $\text{(5.656854, 135°)}$

9. Cartesian point (x, y): (0, 9)

1. The x-coordinate is the across part of the distance: multiply r by the cosine of θ

   $x$$=9\times \text{cos}90\text{°}$$=0$

2. The y-coordinate is the up part: multiply r by the sine of θ

   $y$$=9\times \text{sin}90\text{°}$$=9$

3. Putting x and y together, the Cartesian point is

   $\text{(0, 9)}$

10. Cartesian point (x, y): (0, 10)

1. The x-coordinate is the across part of the distance: multiply r by the cosine of θ

   $x$$=10\times \text{cos}90\text{°}$$=0$

2. The y-coordinate is the up part: multiply r by the sine of θ

   $y$$=10\times \text{sin}90\text{°}$$=10$

3. Putting x and y together, the Cartesian point is

   $\text{(0, 10)}$