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).