Polar coordinates describe a point by its distance from the origin, ( r ), and its angle from the positive x-axis, ( theta ). To convert, use ( x = rcostheta ), ( y = rsintheta ). Conversely, use ( r^2 = x^2 + y^2 ), ( tantheta = frac{y}{x} ). A polar equation ( r = f(theta) ) gives r for each angle; the area it encloses is ( A = frac12int_alpha^beta r^2,dtheta ).
Rectangular coordinates ( (x, y) ) locate a point by going across and up. Polar coordinates ( (r, theta) ) locate it by turning and going out. Circles, spirals, petals and orbits that are awkward in x and y often have very simple polar equations. As a result, polar coordinates appear in precalculus, Calculus 2, physics and engineering.
Polar and rectangular conversions
| To find | Formula |
|---|---|
| x from ( (r, theta) ) | ( x = rcostheta ) |
| y from ( (r, theta) ) | ( y = rsintheta ) |
| r from ( (x, y) ) | ( r = sqrt{x^2 + y^2} ) |
| ( theta ) from ( (x, y) ) | ( tantheta = frac{y}{x} ), choosing the quadrant of ( (x, y) ) |
These come straight from the right triangle formed by the point, the origin and the x-axis: r is the hypotenuse (see trigonometric functions and the unit circle).
Example 1: polar to rectangular
( (3, frac{pi}{3}) ): ( x = 3cosfrac{pi}{3} = frac32 ) and ( y = 3sinfrac{pi}{3} = frac{3sqrt3}{2} approx 2.598 ). So the point is ( left( frac32, frac{3sqrt3}{2} right) ).
Example 2: rectangular to polar
( (1, 1) ): ( r = sqrt2 ) and ( tantheta = 1 ), with the point in Quadrant I, so ( theta = frac{pi}{4} ). The point is ( left( sqrt2, frac{pi}{4} right) ).
Example 3: watch the quadrant
( (-2, 2sqrt3) ): ( r = sqrt{4 + 12} = 4 ). ( tantheta = -sqrt3 ), and a calculator’s arctan gives ( -frac{pi}{3} ), which is in Quadrant IV. The point is in Quadrant II, so ( theta = pi – frac{pi}{3} = frac{2pi}{3} ). Answer: ( left( 4, frac{2pi}{3} right) ). This quadrant check is the most common source of errors (see inverse trig functions).
One point, many names
Unlike rectangular coordinates, a point has infinitely many polar names. ( (2, frac{pi}{6}) ), ( (2, frac{13pi}{6}) ) and ( (-2, frac{7pi}{6}) ) are the same point; a negative r means “go the opposite way”.
Converting equations
Rectangular to polar: substitute ( x = rcostheta ), ( y = rsintheta ).
- The circle ( x^2 + y^2 = 4x ) becomes ( r^2 = 4rcostheta ), so ( r = 4costheta ).
- The vertical line ( x = 3 ) becomes ( rcostheta = 3 ), so ( r = 3sectheta ).
Polar to rectangular: multiply through by r if that creates ( rcostheta ), ( rsintheta ) or ( r^2 ).
- ( r = 2sintheta ) becomes ( r^2 = 2rsintheta ), so ( x^2 + y^2 = 2y ): a circle of radius 1 centered at ( (0, 1) ).
Graphs of polar equations
| Equation | Graph |
|---|---|
| ( r = a ) | circle of radius ( a ) centered at the origin |
| ( theta = alpha ) | line through the origin at angle ( alpha ) |
| ( r = 2acostheta ) | circle of radius ( a ) through the origin, centered on the x-axis |
| ( r = 2asintheta ) | circle of radius ( a ) through the origin, centered on the y-axis |
| ( r = a(1 pm costheta) ) | cardioid (heart shape) |
| ( r = a + bcostheta ) | limaçon (with an inner loop if ( a < b )) |
| ( r = acos(ntheta) ) | rose: n petals if n is odd, 2n petals if n is even |
| ( r^2 = a^2cos 2theta ) | lemniscate (figure eight) |
| ( r = atheta ) | spiral of Archimedes |
How to sketch a polar graph
- Make a table of ( theta ) and r at the special angles ( 0, frac{pi}{6}, frac{pi}{4}, frac{pi}{3}, frac{pi}{2}, dots )
- Plot each point by turning to ( theta ) and going out r.
- Use symmetry: if replacing ( theta ) by ( -theta ) leaves the equation unchanged (as with cosine), the graph is symmetric about the x-axis.
- Find where ( r = 0 ): the graph passes through the origin there.
Example 4: the cardioid ( r = 1 + costheta ). At ( theta = 0 ), ( r = 2 ); at ( frac{pi}{2} ), ( r = 1 ); at ( pi ), ( r = 0 ) (the cusp at the origin); at ( frac{3pi}{2} ), ( r = 1 ). Also, it’s symmetric about the x-axis.
Slope of a polar curve
Treat ( x = r(theta)costheta ) and ( y = r(theta)sintheta ) as parametric equations in ( theta ). By the chain rule and product rule:
$$frac{dy}{dx} = frac{dy/dtheta}{dx/dtheta} = frac{r'sintheta + rcostheta}{r'costheta - rsintheta}$$
Example 5. For ( r = 1 + costheta ) at ( theta = frac{pi}{2} ): ( r = 1 ) and ( r’ = -sintheta = -1 ).
$$frac{dy}{dx} = frac{(-1)(1) + (1)(0)}{(-1)(0) - (1)(1)} = frac{-1}{-1} = 1$$
The tangent line at the point ( (0, 1) ) is ( y = x + 1 ) (see the equation of a tangent line). Horizontal tangents occur where ( frac{dy}{dtheta} = 0 ). Vertical tangents, however, occur where ( frac{dx}{dtheta} = 0 ).
Area in polar coordinates
Polar regions are swept out like slices of pie. A thin slice with angle ( dtheta ) is nearly a sector of radius r, with area ( frac12r^2,dtheta ). Adding the slices gives
$$A = frac12int_alpha^beta r^2,dtheta = frac12int_alpha^beta [f(theta)]^2,dtheta$$
Example 6: area of the cardioid.
$$A = frac12int_0^{2pi}(1 + costheta)^2,dtheta = frac12int_0^{2pi}left(1 + 2costheta + frac{1 + cos 2theta}{2}right)dtheta = frac12cdot 3pi = frac{3pi}{2}$$
The step that makes ( cos^2theta ) integrable is the power-reducing identity (see trig identities and the integral of sin²x). Check the value in the calculator below, which evaluates ( int_0^{2pi}frac12(1 + cos x)^2,dx ) with x standing for ( theta ):
The answer is ( 4.712389 approx frac{3pi}{2} ).
Example 7: a circle as a check. ( r = 2sintheta ) traces its circle once for ( 0 le theta le pi ): ( A = frac12int_0^pi 4sin^2theta,dtheta = pi ), the area of a circle of radius 1, as it should be.
Area between two polar curves: ( A = frac12int_alpha^beta (r_{text{outer}}^2 – r_{text{inner}}^2),dtheta ), with ( alpha ) and ( beta ) found where the curves meet. Note it’s the difference of the squares, not the square of the difference.
Arc length in polar coordinates
$$L = int_alpha^betasqrt{r^2 + left(frac{dr}{dtheta}right)^2},dtheta$$
For the cardioid ( r = 1 + costheta ), ( r^2 + r’^2 = 2 + 2costheta = 4cos^2frac{theta}{2} ). As a result, the total length works out to exactly 8. Compare this to the rectangular arc length formula.
Where polar coordinates are used
- Physics and astronomy: planetary orbits are conic sections with simple polar equations, ( r = frac{ed}{1 + ecostheta} ).
- Engineering: antenna radiation patterns and microphone pickup patterns (the “cardioid microphone”).
- Navigation: radar and sonar report distance and bearing, a polar description.
- Multivariable calculus: double integrals over circles and rings are far easier in polar form, with ( dA = r,dr,dtheta ) (see double integrals and multivariable calculus).
Common mistakes
- Wrong quadrant for ( theta ). Always check where ( (x, y) ) is before trusting arctan.
- Forgetting the ( frac12 ) and the square in the area formula.
- Integrating over too wide an interval. The equation ( r = 2sintheta ) traces its circle once on ( [0, pi] ); however, integrating to ( 2pi ) doubles the area.
- Squaring the difference for area between curves instead of subtracting squares.
- Degrees in calculus. Polar calculus formulas assume ( theta ) in radians (see degrees to radians).
Practice problems
- Convert ( (4, frac{5pi}{6}) ) to rectangular coordinates.
- Convert ( (0, -3) ) to polar coordinates with ( r > 0 ).
- Convert ( x^2 + y^2 = 9 ) to a polar equation.
- Convert ( r = 6costheta ) to rectangular form and describe the graph.
- How many petals does ( r = sin(3theta) ) have? And ( r = cos(2theta) )?
- Find the area enclosed by ( r = 3 ).
Answers: (1) ( (-2sqrt3, 2) ); (2) ( (3, frac{3pi}{2}) ); (3) ( r = 3 ); (4) ( (x – 3)^2 + y^2 = 9 ), a circle of radius 3 centered at ( (3, 0) ); (5) 3 petals; 4 petals; (6) ( 9pi ).
FAQ
What are polar coordinates?
A way to locate a point by its distance r from the origin and its angle ( theta ) from the positive x-axis, written ( (r, theta) ).
How do you convert polar to rectangular coordinates?
Use ( x = rcostheta ) and ( y = rsintheta ).
How do you convert rectangular to polar coordinates?
Use ( r = sqrt{x^2 + y^2} ) and ( tantheta = frac{y}{x} ), adjusting ( theta ) to the correct quadrant.
What is the formula for area in polar coordinates?
( A = frac12int_alpha^beta r^2,dtheta ).
What is a polar equation?
An equation relating r and ( theta ), usually written ( r = f(theta) ). For example, ( r = 1 + costheta ) is a cardioid and ( r = 2 ) is a circle.
Further reading
- Polar coordinate system (Wikipedia) — conversions, common polar curves and calculus in polar form.
- Calculus II (Paul’s Online Math Notes) — the Parametric Equations and Polar Coordinates chapter, with tangents, area and arc length.
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