Polar Coordinates and Polar Curves
Locate a point by distance and angle instead of two displacements, convert between the two systems, and sketch curves that are simpler in polar form.
What a learner can do afterwards
- Convert a point and an equation between Cartesian and polar form
- Sketch a circle, a spiral and a rose curve from a polar equation by tracking the radius as the angle turns
- Say which shapes gain from polar form and which lose, with a reason
1 · Read
Polar coordinates are written (r, theta). Measure theta from the positive x-axis. Anticlockwise angles are positive; clockwise angles are negative. Use radians unless degrees are stated. A full turn is 2pi radians, or 360 degrees. For r > 0, move r units along the angle's ray. The unit-circle coordinates give x = r cos theta and y = r sin theta.
For the polar point (4, pi/2), pi/2 is 90 degrees. Calculate x = 4 cos(pi/2) = 0 and y = 4 sin(pi/2) = 4. Its Cartesian coordinates are (0, 4). For (7, pi), pi is 180 degrees: cosine is -1 and sine is 0. The Cartesian coordinates are (-7, 0).
These calculations give the two Cartesian coordinates. The cells do not draw the point.
Use r = sqrt(x² + y²), the nonnegative distance. For non-origin points, find theta from the quadrant and tan theta = y/x (x nonzero). Equal nonzero absolute coordinates give a 45-degree reference angle. Read axis directions directly. Choose an angle from zero up to, but not including, one full turn. At the origin, r = 0; any angle works.
Negative r moves |r| units opposite theta; the coordinate formulas still work. Equivalent names either keep r and add a full turn, or reverse r's sign and add 180 degrees (pi radians).
Sketch r = f(theta) by calculating radii, plotting pairs and joining in angle order. For r > 0, distance is r; the rays 0, pi/2, pi, 3pi/2, 2pi point right, up, left, down, right. Constant positive r traces an origin-centred circle. The spiral r = k theta (k > 0, theta >= 0, radians) grows outward: a full turn adds 2pi to theta and 2pi k to r.
For r = a cos(n theta), use a > 0 and integer n >= 2. Sample angles where n theta is a multiple of pi/2: cosine cycles 1, 0, -1, 0. Multiply by a for r. Plot negative r on the opposite ray; at theta = pi/2, r = -a plots down. Join in angle order. Zeros return to the origin between tips. Each tip is distance a away. Odd n gives n petals; even n gives 2n. Retracing adds no petals. The first positive zero has n theta = pi/2.
For r = 3 cos(3 theta), one petal runs from theta = -pi/6 to pi/6. Its radius grows from 0 to 3 at theta = 0, then returns to 0. The other petals point at 2pi/3 and 4pi/3. At theta = pi/3, r = -3: plot 3 units along the opposite ray, at direction 4pi/3. This reaches an existing petal tip, not a fourth petal.
Convert equations using x = r cos theta, y = r sin theta and x² + y² = r². For x² + y² = 49, substitute r² to get r² = 49. With r >= 0, use r = 7 and let theta make a full turn. If signed radii are allowed, r = -7 traces the same circle. Conversion also works the other way: replace r sin theta with y, or r cos theta with x.
Compare equations for the same curve. An origin-centred circle has the short polar form r = a. A vertical line x = 5 becomes r cos theta = 5, or r = 5/cos theta where cosine is not zero. This is also written r = 5 sec theta: sec means 1/cos. Cartesian form fixes x directly. A fixed polar angle gives a ray when r >= 0 and a whole line when signed r is unrestricted. Check restrictions when comparing forms.
Walk the angle, step out the radius, and pick the system that makes your curve look simplest.
Lesson and practice text written for this dataset. OpenStax polar-coordinate sections were consulted for mathematical facts and teaching structure; their prose was not reused.
Readings, questions and flashcards written for this dataset. OpenStax polar-coordinate sections and WeBWorK problem records were studied for facts and question design; their prose was not reused.
More to read
Checking two names for one point
Imagine a point 5 units from the origin on the ray at -30 degrees. The polar name (5, 330 degrees) reaches it too: the two angles differ by a full turn.
Now consider (-5, 150 degrees). The 150-degree ray points opposite the original ray. The negative radial coordinate reverses the movement along that ray, so this pair also reaches the original point. Both changes matter. Changing only 5 to -5 would reach the opposite point.
You can check polar names with x = r cos theta and y = r sin theta. Equivalent names must give the same x and the same y. A sign change in r is balanced by a half-turn in theta because both sine and cosine then change sign.
The origin is a special case. When r = 0, both Cartesian coordinates are zero for every angle. Its many polar names all refer to the same point, whatever direction the angle gives.
Planning a polar sketch
Start a sketch by choosing an angle interval. Calculate several angle-radius pairs, plot them on a polar grid, and connect nearby points in angle order. More sample angles help where the curve bends or passes through the origin.
A constant positive radius returns to its starting point after one turn and traces a circle centred at the origin. For r = 2 theta with theta >= 0 in radians, the radii at 0, pi and 2pi are 0, 2pi and 4pi. After one full turn, the ray points in the same direction but the point is farther out. Each later turn adds another 4pi to the radius.
For r = 3 cos(3 theta), begin with theta from -pi/6 to pi/6. The radius is zero at both ends and reaches 3 in between. These values trace one petal. Two more petals point at 2pi/3 and 4pi/3.
Keep negative radii when filling the table. At theta = pi/3, the radius is -3, so the point lies 3 units along the opposite ray. It reaches a petal already counted. Counting visits to tips would therefore overcount the petals.
Keep the points when changing equations
An equation conversion should keep the same geometric points. Substitute x = r cos theta, y = r sin theta and x² + y² = r², then check any restrictions.
For a circle centred at the origin, a nonnegative constant r gives its radius directly. Squaring that constant gives the right side of its Cartesian equation x² + y² = r². A negative constant radial coordinate traces the same circle when the angle is unrestricted.
A line may have a shorter Cartesian description. Substituting into x = 5 gives r cos theta = 5. Writing r = 5/cos theta requires cos theta not to be zero. The original equation fixes x without that trigonometric expression. Conversely, r sin theta = c becomes y = c: a horizontal line.
A fixed angle needs a radial restriction too. With r >= 0 it gives a ray from the origin. Allowing every signed r extends that ray through the origin into a whole line.
When finding a point's polar angle, tangent alone also needs a check. Opposite quadrants can have the same tangent. Use the signs of x and y to select the quadrant, and handle axis points directly.
2 · Watch
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