--- title: "Polar Coordinates and Polar Curves" description: "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." canonical: https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v source: https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Geometry · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/polar-coordinates-and-polar-curves-mt_R5QMUJvA1v ## Ready when they can - 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 ## Lesson: Finding points by distance and angle 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. **Example.** 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). 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. *(drawing: For r = 2 theta, each quarter-turn reaches farther out. Use these pairs to start a sketch on a polar grid.)* 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. **Example.** 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. *(drawing: The first three pairs trace one petal. The negative radius in the next pair must be plotted on the opposite ray.)* 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. **Recap.** 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.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4) - [The Unit Circle and the Graphs of Sine, Cosine and Tangent](https://lightmysky.com/learn/mathematics/the-unit-circle-and-the-graphs-of-sine-cosine-and-tangent-mt_VJaimevi3-) ## Optional connections - [Conic Sections: Ellipse, Parabola and Hyperbola](https://lightmysky.com/learn/mathematics/conic-sections-ellipse-parabola-and-hyperbola-mt_V4YSI3a_Eu)