Unit 14: Polar Coordinates
Locating a point by a distance and an angle instead of two perpendicular distances, and watching curves bloom.
6 topics · 25 questions
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Converting Between Polar and Rectangular Coordinates
- How it worksHow do x = r cos(theta) and y = r sin(theta) translate a polar point into rectangular coordinates and back?
- Worked exampleFind the rectangular coordinates of the polar point (4, π/3).
- Worked exampleFind polar coordinates for the rectangular point (-1, 1).
- Common mistakeWhy doesn't every point have just one correct pair of polar coordinates?
- Going furtherWhat happens to a rectangular equation like x2 + y2 = 9 when it's rewritten entirely in polar coordinates?
Your turn1 of 2
Find the x-coordinate of the rectangular point corresponding to the polar point (r, theta), as a function of r and theta.Graphing a Polar Curve as Theta Sweeps
- How it worksHow does a point trace out the full curve of r = 1 + cos(theta) as theta sweeps from 0 to 2 π?
- How it worksHow does r shrinking to zero and growing again as theta sweeps create the inner loop of r = 1 + 2cos(theta)?
- Worked exampleFind the points where the curve r = 2 sin(theta) crosses the pole as theta sweeps from 0 to 2 π.
- Common mistakeWhy doesn't a negative value of r just get skipped when plotting a polar curve?
- Going furtherWhat happens to the curve r = cos(2 theta) that makes it trace out four petals instead of one?
Your turn1 of 2
For the curve r = 4 + cos(theta), find r at theta = 0.95.Cardioids, Roses, and Limacons
- How it worksHow does the ratio between a and b in r = a + b cos(theta) decide whether the curve is a cardioid or a limacon?
- Worked exampleFind how many petals the rose r = 3 sin(4 theta) has.
- Worked exampleFind how many petals the rose r = 2 cos(5 theta) has.
- Common mistakeWhy doesn't r = cos(n theta) always produce n petals regardless of whether n is even or odd?
Your turn1 of 2
How many petals does the rose r = 3 sin(4 theta) have?Tangent Lines to a Polar Curve
- How it worksHow does writing x = r(theta) cos(theta) and y = r(theta) sin(theta) turn a polar curve into a parametric one?
- Worked exampleFind the slope of the tangent line to r = 1 + sin(theta) at theta = π/6.
- Common mistakeWhy isn't the slope of a tangent line to a polar curve just dr/d(theta)?
Your turn1 of 2
Find the slope of the tangent line to r = 3 + sin(theta), as a function of theta.Area Enclosed by a Polar Curve
- How it worksHow does integrating one-half r squared d theta add up a fan of thin wedges into the total enclosed area?
- Worked exampleFind the area enclosed by one petal of the rose r = cos(2 theta).
- Worked exampleFind the area inside the cardioid r = 1 + cos(theta).
- Common mistakeWhy doesn't the polar area formula just integrate r d theta the way arc length integrates a length element?
Your turn1 of 2
Find the area enclosed by one petal of the rose r = 3cos(4 theta).Arc Length of a Polar Curve
- How it worksHow does treating a polar curve as parametric in theta turn its length into an integral of √(r2 + (dr/d theta)2)?
- Worked exampleFind the arc length of the cardioid r = 1 + cos(theta) for theta from 0 to π.
- Common mistakeWhy doesn't the polar arc length formula just use r as the only thing under the square root?
- Going furtherWhat happens to the polar arc length integral when r is constant, tracing a plain circle?
Your turn1 of 2
For the cardioid r = 4(1 + cos(theta)), the arc-length integral is the integral of √(r2 + (dr/d theta)2) d theta. Write this integrand as a function of theta.