Converting Between Polar and Rectangular Coordinates | Precalculus

Converting Polar to Rectangular Coordinates

• 💡 To convert from polar coordinates (r, (\theta)) to rectangular coordinates (x, y), use the formulas: x = r cos((\theta)) and y = r sin((\theta)).
• 🎯 For example, given polar coordinates with r = 7 and (\theta) = 3(\pi)/5, the rectangular coordinates are x = 7 cos(3(\pi)/5) (\approx) -2.163 and y = 7 sin(3(\pi)/5) (\approx) 6.657.
• 📈 Calculators should be in radian mode for these calculations.

Converting Rectangular to Polar Coordinates

• 🔑 To convert from rectangular coordinates (x, y) to polar coordinates (r, (\theta)), use the formulas: r = (\sqrt{x^2 + y^2}) and (\theta) = arctan(y/x).
• ⚠️ Be cautious with the arctan function, as it typically returns angles between -90° and 90°. This may require adding or subtracting 180° (or (\pi) radians) to find the correct angle based on the quadrant of the (x, y) point.
• 📍 For instance, for rectangular coordinates (-2, 11), which lie in the third quadrant:
  • Calculate r: r = (\sqrt{(-2)^2 + 11^2}) = (\sqrt{4 + 121}) = (\sqrt{125}) (\approx) 11.180.
  • Calculate (\theta): First, find the reference angle using arctan(11/-2) (\approx) -79.7°. Since the point is in the third quadrant, add 180° to get the correct angle: (\theta) (\approx) -79.7° + 180° = 100.3°.
  • The approximate polar coordinates are (11.18, 100.3°).
• 📐 When working in degrees, ensure the calculator is set to degree mode for angle calculations.