Glossary

Angle (θ)

Criticality: 3

In polar coordinates, the directed angle measured counterclockwise from the positive x-axis (polar axis) to the line segment connecting the origin to a point.

Example:

When sketching a rose curve, you typically sweep the angle from 0 to 2π to trace all petals.

Area of a Sector

Criticality: 2

The area of a 'pizza slice' of a circle in polar coordinates, given by the formula (1/2)r²θ, where r is the radius and θ is the angle in radians.

Example:

A sector with a radius of 6 and an angle of π/4 radians has an area of (1/2)(6²)(π/4) = 9π/2 square units.

Cartesian Coordinates

Criticality: 1

A system for locating points in a plane using horizontal (x) and vertical (y) distances from the origin.

Example:

The point (5, 12) in (Cartesian coordinates) is found by moving 5 units right and 12 units up from the origin.

Definite Integral for Polar Area

Criticality: 3

The formula $A = \frac{1}{2}\int_{a}^{b}[f(\theta)]^2d\theta$ used to calculate the area enclosed by a polar curve $r=f(\theta)$ from angle $a$ to angle $b$.

Example:

To find the area of a cardioid, you would apply the definite integral for polar area over the appropriate angular range, typically 0 to 2π.

Limacon

Criticality: 1

A family of polar curves defined by equations of the form $r = a \pm b \sin(\theta)$ or $r = a \pm b \cos(\theta)$, which can exhibit various shapes including cardioids or inner loops.

Example:

The curve $r = 1 + 2cos(\theta)$ is a limacon with an inner loop, requiring careful consideration of the integration limits for its area.

Polar Coordinates

Criticality: 3

A system for locating points in a plane using a distance from the origin (radius, r) and an angle from the positive x-axis (theta, θ).

Example:

To plot the point (polar coordinates) (3, π/6), you move 3 units from the origin along a ray at a 30-degree angle.

Polar Curve

Criticality: 3

A curve whose points are defined by a polar equation, typically expressed in the form r = f(θ).

Example:

The equation $r = 2 + 2cos(\theta)$ describes a polar curve known as a cardioid.

Radius (r)

Criticality: 3

In polar coordinates, the directed distance from the origin (pole) to a specific point on a curve.

Example:

For the polar curve r = 4cos(θ), the radius changes as the angle θ sweeps, tracing out a circle.

Symmetry (in polar curves)

Criticality: 2

A property of a polar curve where a portion can be reflected or rotated to match another, often used to simplify area calculations by integrating over a smaller interval and multiplying.

Example:

For a rose curve, recognizing its symmetry allows you to calculate the area of just one petal and then multiply by the total number of petals to find the total area.

Trigonometric Identities

Criticality: 2

Equations involving trigonometric functions that are true for all valid variable values, often used to simplify expressions or integrals.

Example:

When integrating $cos^2(\theta)$ , you often use the trigonometric identity $cos^2(\theta) = \frac{1+cos(2\theta)}{2}$ to make the integration straightforward.

Glossary

Angle (θ)

Criticality: 3

In polar coordinates, the directed angle measured counterclockwise from the positive x-axis (polar axis) to the line segment connecting the origin to a point.

Example:

When sketching a rose curve, you typically sweep the angle from 0 to 2π to trace all petals.

Area of a Sector

Criticality: 2

The area of a 'pizza slice' of a circle in polar coordinates, given by the formula (1/2)r²θ, where r is the radius and θ is the angle in radians.

Example:

A sector with a radius of 6 and an angle of π/4 radians has an area of (1/2)(6²)(π/4) = 9π/2 square units.

Cartesian Coordinates

Criticality: 1

A system for locating points in a plane using horizontal (x) and vertical (y) distances from the origin.

Example:

The point (5, 12) in (Cartesian coordinates) is found by moving 5 units right and 12 units up from the origin.

Definite Integral for Polar Area

Criticality: 3

The formula $A = \frac{1}{2}\int_{a}^{b}[f(\theta)]^2d\theta$ used to calculate the area enclosed by a polar curve $r=f(\theta)$ from angle $a$ to angle $b$.

Example:

To find the area of a cardioid, you would apply the definite integral for polar area over the appropriate angular range, typically 0 to 2π.

Limacon

Criticality: 1

A family of polar curves defined by equations of the form $r = a \pm b \sin(\theta)$ or $r = a \pm b \cos(\theta)$, which can exhibit various shapes including cardioids or inner loops.

Example:

The curve $r = 1 + 2cos(\theta)$ is a limacon with an inner loop, requiring careful consideration of the integration limits for its area.

Polar Coordinates

Criticality: 3

A system for locating points in a plane using a distance from the origin (radius, r) and an angle from the positive x-axis (theta, θ).

Example:

To plot the point (polar coordinates) (3, π/6), you move 3 units from the origin along a ray at a 30-degree angle.

Polar Curve

Criticality: 3

A curve whose points are defined by a polar equation, typically expressed in the form r = f(θ).

Example:

The equation $r = 2 + 2cos(\theta)$ describes a polar curve known as a cardioid.

Radius (r)

Criticality: 3

In polar coordinates, the directed distance from the origin (pole) to a specific point on a curve.

Example:

For the polar curve r = 4cos(θ), the radius changes as the angle θ sweeps, tracing out a circle.

Symmetry (in polar curves)

Criticality: 2

A property of a polar curve where a portion can be reflected or rotated to match another, often used to simplify area calculations by integrating over a smaller interval and multiplying.

Example:

For a rose curve, recognizing its symmetry allows you to calculate the area of just one petal and then multiply by the total number of petals to find the total area.

Trigonometric Identities

Criticality: 2

Equations involving trigonometric functions that are true for all valid variable values, often used to simplify expressions or integrals.

Example:

When integrating $cos^2(\theta)$ , you often use the trigonometric identity $cos^2(\theta) = \frac{1+cos(2\theta)}{2}$ to make the integration straightforward.