# Polar to Rectangular: Formula & Worked Examples TL;DR To convert polar to rectangular coordinates, use x = r cos θ and y = r sin θ, turning a distance-and-angle pair (r, θ) into an across-and-up pair (x, y). This guide derives the formulas from a right triangle, works through examples in every quadrant, and lists the common mistakes. ## What Does Polar to Rectangular Mean? Converting polar to rectangular means rewriting a point given as (r,θ), a distance r from the origin and an angle θ from the positive x-axis, as a point (x,y) on the usual grid. The two conversion formulas are: x = r cos θ y = r sin θ A **rectangular** (or Cartesian) coordinate locates a point by how far across and how far up it sits, the familiar (x,y) of the coordinate plane. A **polar** coordinate locates the same point by how far away it is (r) and in what direction (θ). Both name the same point; they just use different instructions to get there. The key idea: the **distance r is a hypotenuse, and x and y are its two legs.** That triangle is where the formulas come from. ## Where The Formulas Come From The conversion is not a rule to memorise blindly; it is the definition of cosine and sine. Drop a perpendicular from the point P down to the x-axis. You get a right triangle with: - the hypotenuse equal to r (the distance from the origin to P), - the horizontal leg equal to x, - the vertical leg equal to y, - the angle θ at the origin. In any right triangle, cosine is the adjacent leg over the hypotenuse, and sine is the opposite leg over the hypotenuse: cos θ = x/r sin θ = y/r Multiply each equation by r to solve for the rectangular coordinate: x = r cos θ y = r sin θ That is the entire derivation. The angle decides the direction; the radius scales it to the right distance. **How do you handle an angle that is not in the first quadrant?** Use the angle to find the quadrant, then let the signs of cosine and sine do the work. The formulas x = r cos θ and y = r sin θ automatically produce the right signs once you use the correct cosine and sine values for that angle. ## Examples of Polar to Rectangular Conversion These build from a clean first-quadrant point to angles in other quadrants and a negative radius. Each problem statement is bold; the steps are plain. ### Example 1 **Convert the polar point (4, 60°) to rectangular coordinates.** Use the formulas with r = 4 and θ = 60°: x = 4 cos 60° = 4 ⋅ 1/2 = 2 y = 4 sin 60° = 4 ⋅ √3/2 = 2√3 Final answer: (2, 2√3), roughly (2, 3.46). ### Example 2 **Convert the polar point (5, 90°) to rectangular coordinates.** Your first instinct may be to expect both x and y to be nonzero because r = 5 is a real distance. Let's compute: x = 5 cos 90° = 5 ⋅ 0 = 0 y = 5 sin 90° = 5 ⋅ 1 = 5 Final answer: (0, 5). ### Example 3 **Convert the polar point (6, 120°) to rectangular coordinates.** The angle 120° lands in the second quadrant, where cosine is negative and sine is positive: x = 6 cos 120° = 6 ⋅ (-1/2) = -3 y = 6 sin 120° = 6 ⋅ √3/2 = 3√3 Final answer: (-3, 3√3), roughly (-3, 5.20). ### Example 4 **Convert the polar point (4, π/3) to rectangular coordinates.** The angle is in radians: π/3 = 60°. Same as Example 1 in disguise: x = 4 cos π/3 = 4 ⋅ 1/2 = 2 y = 4 sin π/3 = 4 ⋅ √3/2 = 2√3 Final answer: (2, 2√3). ### Example 5 **Convert the polar point (3, 210°) to rectangular coordinates.** The angle 210° is in the third quadrant, where both cosine and sine are negative: x = 3 cos 210° = 3 ⋅ (-√3/2) = -3√3/2 y = 3 sin 210° = 3 ⋅ (-1/2) = -3/2 Final answer: (-3√3/2, -3/2). ### Example 6 **A radar station tracks a boat 8 km away at a bearing that corresponds to a polar angle of 300°. Where is the boat in rectangular (east, north) coordinates?** Take r = 8 and θ = 300°, which is in the fourth quadrant (cosine positive, sine negative): x = 8 cos 300° = 8 ⋅ 1/2 = 4 y = 8 sin 300° = 8 ⋅ (-√3/2) = -4√3 Final answer: (4, -4√3), roughly (4, -6.93). ## Why This Conversion Matters: "Two Ways to Name the Same Place" Some quantities arrive naturally as a distance and a direction: a radar ping, a robot arm's reach, a wind reading, a force pushing at an angle. Others are easier as across and up: plotting on a screen, adding two movements, programming a grid. Polar to rectangular is the bridge that lets you take a measurement in the first language and do arithmetic in the second. - **Adding directions.** You cannot simply add two polar pairs, but you can add their rectangular components. Convert each to (x,y), add the x's and the y's, and you have the combined result. - **Plotting on a grid.** Screens, maps, and graph paper are rectangular. A point known only by distance and angle has to be converted before it can be drawn. - **Engineering and signals.** Alternating-current voltages, complex numbers, and rotating machinery are often described by magnitude and angle, then converted to components to be combined. ## Common Mistakes With Polar to Rectangular ### Mistake 1: Degrees-versus-radians mix-up **Where it slips in:** Plugging a degree value into a calculator set to radians, or the reverse. **Don't do this:** Computing cos 60° with the calculator in radian mode and getting roughly −0.95 instead of 0.5. **The correct way:** Check the angle's units first. ### Mistake 2: Ignoring the quadrant signs **Where it slips in:** Treating cosine and sine as always positive. **Don't do this:** Writing cos 120° = 1/2 (positive) when 120° is in the second quadrant, where cosine is negative. **The correct way:** Use the angle to find the quadrant, then apply the correct signs. ### Mistake 3: Swapping the formulas for x and y **Where it slips in:** Pairing sine with x and cosine with y. **Don't do this:** Writing x = r sin θ and y = r cos θ. **The correct way:** Cosine goes with x and sine goes with y. ## Conclusion - **Polar to rectangular** conversion turns (r,θ) into (x,y) using x = r cos θ and y = r sin θ. - The formulas come from a right triangle where r is the hypotenuse and x, y are the legs. - The angle's quadrant sets the signs of x and y, so quadrant checks matter. - Match calculator mode to the angle's units, degrees or radians, before computing. - Cosine always pairs with x; sine always pairs with y. ## A Practical Next Step Practise these to lock it in: convert (2, 45°); convert (10, 270°); and convert (6, 2π/3).