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.

Use the relations $x = r\cos\theta$, $y = r\sin\theta$, and $r = \sqrt{x^2 + y^2}$ to eliminate r and θ and express the equation in terms of x and y.

Find $\frac{dr}{d\theta}$, set it equal to zero, solve for $\theta$, and plug the $\theta$ values back into the original equation to find the corresponding r values. Also, check endpoints.

It allows for easier visualization and graphing on a traditional x-y coordinate plane, especially for complex functions.

It represents the radial component of the curve and indicates the instantaneous rate of change of the distance from the origin.

It represents the radial curvature of the curve and indicates the rate of change of the curvature.

Polar equations can be expressed parametrically using $x = r\cos\theta$ and $y = r\sin\theta$, where $\theta$ is the parameter.

It gives the slope of the tangent line to the curve at a given point in Cartesian coordinates, indicating the direction of the curve at that point.

Because y and x are both functions of r and θ, and r is a function of θ, the chain rule is needed to relate the derivatives.

Calculate $\frac{dy}{dx}$ at the given angle, find the x and y coordinates using $x = r\cos\theta$ and $y = r\sin\theta$, then use the point-slope form to find the equation of the tangent line.

Trigonometric identities are used to simplify expressions, convert between polar and Cartesian coordinates, and solve equations involving trigonometric functions of θ.