A two-dimensional coordinate system defined by a distance (r) from the origin and an angle (θ) from the positive x-axis.

Functions graphed in a polar coordinate system using distance (r) from a fixed point (pole) and angle (θ) from the positive x-axis.

A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

The rate of change of the distance from the origin with respect to the angle θ. Helps find points furthest or closest to the origin.

The slope of the tangent line to the polar curve in Cartesian coordinates.

The fixed point from which the distance 'r' is measured in a polar coordinate system. It is essentially the origin.

The first derivative of r with respect to θ, denoted as r'(θ), representing the instantaneous rate of change of the distance from the origin.

The second derivative of r with respect to θ, denoted as r''(θ), representing the rate of change of the curvature.

The angle measured counter-clockwise from the positive x-axis to the line connecting the pole to the point.

Equations expressed in terms of x and y coordinates, representing relationships between these variables on a Cartesian plane.

Polar to Cartesian: Uses $x = r\cos\theta$ and $y = r\sin\theta$ to eliminate r and θ. | Cartesian to Polar: Uses $r = \sqrt{x^2 + y^2}$ and $\theta = \arctan(\frac{y}{x})$ to eliminate x and y.

$\frac{dr}{d\theta}$: Gives the rate of change of the distance from the origin. | $\frac{dy}{dx}$: Gives the slope of the tangent line in Cartesian coordinates.

$r = a\cos(\theta)$: Circle centered on the x-axis. | $r = a\sin(\theta)$: Circle centered on the y-axis.

With Inner Loop: |a| < |b| in $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$. | Without Inner Loop: |a| >= |b| in $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$.

Polar: Uses distance from origin (r) and angle (θ). | Cartesian: Uses horizontal (x) and vertical (y) distances from axes.

$\frac{dr}{d\theta}$: Radial component, rate of change of distance from the origin. | $\frac{dy}{dx}$: Slope of the tangent line in Cartesian coordinates.

$\frac{dr}{d\theta}$ finds where the curve is furthest or closest to the origin. | $\frac{dy}{dx}$ finds the slope of the tangent line at a point on the curve.

n is even: Rose curve with 2n petals. | n is odd: Rose curve with n petals.

Sine: Often associated with vertical symmetry or curves aligned along the y-axis. | Cosine: Often associated with horizontal symmetry or curves aligned along the x-axis.

Both are cardioids. $r = a + a\cos(\theta)$: Symmetric about the x-axis. | $r = a + a\sin(\theta)$: Symmetric about the y-axis.

Multiply both sides by r: $r^2 = 2r\sin\theta$. Substitute $r^2 = x^2 + y^2$ and $y = r\sin\theta$: $x^2 + y^2 = 2y$. Complete the square: $x^2 + (y-1)^2 = 1$.

Find x and y: $x = r\cos\theta = \theta\cos\theta$, $y = r\sin\theta = \theta\sin\theta$. Find $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$: $\frac{dx}{d\theta} = \cos\theta - \theta\sin\theta$, $\frac{dy}{d\theta} = \sin\theta + \theta\cos\theta$. Then, $\frac{dy}{dx} = \frac{\sin\theta + \theta\cos\theta}{\cos\theta - \theta\sin\theta}$.

Find $\frac{dr}{d\theta} = -\sin\theta$. Set $- \sin\theta = 0$, so $\theta = 0, \pi$. Evaluate r at these points: $r(0) = 2$, $r(\pi) = 0$. The closest point is 0, the furthest is 2.

1. Find x and y in terms of θ: $x = r\cos(\theta) = \sin(2\theta)\cos(\theta)$, $y = r\sin(\theta) = \sin(2\theta)\sin(\theta)$. 2. Compute $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$. 3. Calculate $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$ at $\theta = \frac{\pi}{4}$. 4. Find the (x, y) coordinates at $\theta = \frac{\pi}{4}$. 5. Use the point-slope form to write the equation of the tangent line.

1. Recall that $r^2 = x^2 + y^2$. 2. Substitute $r^2$ for $x^2 + y^2$ in the given equation. 3. The polar form is $r^2 = 9$, which simplifies to $r = 3$.

1. Find x and y: $x = 2\cos^2(\theta)$, $y = 2\cos(\theta)\sin(\theta) = \sin(2\theta)$. 2. Find $\frac{dx}{d\theta} = -4\cos(\theta)\sin(\theta) = -2\sin(2\theta)$ and $\frac{dy}{d\theta} = 2\cos(2\theta)$. 3. Calculate $\frac{dy}{dx} = \frac{2\cos(2\theta)}{-2\sin(2\theta)} = -\cot(2\theta)$. 4. Evaluate at $\theta = \frac{\pi}{3}$: $\frac{dy}{dx} = -\cot(\frac{2\pi}{3}) = \frac{\sqrt{3}}{3}$.

1. Calculate r: $r = 4\sin(\frac{\pi}{6}) = 4 * \frac{1}{2} = 2$. 2. Use the formula $x = r\cos(\theta)$: $x = 2\cos(\frac{\pi}{6}) = 2 * \frac{\sqrt{3}}{2} = \sqrt{3}$.

1. Calculate r: $r = 2 + \cos(\frac{\pi}{2}) = 2 + 0 = 2$. 2. Use the formula $y = r\sin(\theta)$: $y = 2\sin(\frac{\pi}{2}) = 2 * 1 = 2$.

1. The x-axis corresponds to $y = 0$. 2. Set $y = r\sin(\theta) = 0$. 3. This implies $\sin(\theta) = 0$ (since r is not always zero). 4. Solve for $\theta$: $\theta = n\pi$, where n is an integer.

1. The y-axis corresponds to $x = 0$. 2. Set $x = r\cos(\theta) = 0$. 3. This implies $\cos(\theta) = 0$ (since r is not always zero). 4. Solve for $\theta$: $\theta = \frac{\pi}{2} + n\pi$, where n is an integer.