Cartesian: Uses (x, y), rectangular grid | Polar: Uses (r, θ), circular grid.

y = x: Straight line | r = θ: Spiral.

Cartesian: (x-h)² + (y-k)² = r² | Polar: r = a (centered at origin).

Cartesian: Replace y with -y | Polar: Replace θ with -θ.

Cartesian: Replace x with -x | Polar: Replace θ with π - θ.

r = a cos(θ): Changes diameter of circle | r = cos(aθ): Changes the number of 'petals'.

Cartesian: f(x) = f(x + p) | Polar: f(θ) = f(θ + 2π) (typically).

Cartesian: Unique representation (0,0) | Polar: Infinite representations (0, θ) for any θ.

Cartesian: Straight line | Polar: Requires converting to polar equation, can be more complex.

Cartesian: ∫∫ dxdy | Polar: ∫∫ r dr dθ

A system using (r, θ) to locate points, where r is the distance from the origin and θ is the angle from the positive x-axis.

An equation in the form r = f(θ), where θ is the input angle and r is the output distance from the origin.

A polar function is symmetric about the origin if its graph is unchanged when rotated by 180 degrees.

A polar function is periodic if its graph repeats after a fixed interval of θ.

The distance from the origin (0,0) to the point in the polar plane.

The angle measured counterclockwise from the positive x-axis to the point in the polar plane.

A heart-shaped polar curve, often represented by the equation r = a(1 ± cos θ) or r = a(1 ± sin θ).

A figure-eight-shaped polar curve, often represented by the equation r² = a²cos(2θ) or r² = a²sin(2θ).

It indicates a point located in the opposite direction of the angle θ from the origin.

r = a, where 'a' is the radius of the circle.

1. Create a table of θ and r values. 2. Plot the points (r, θ). 3. Connect the points to form a circle.

1. Test for x-axis symmetry by replacing θ with -θ. 2. Test for y-axis symmetry by replacing θ with π - θ. 3. Test for origin symmetry by replacing r with -r.

1. Use x = r cos(θ) and y = r sin(θ). 2. x = 2 cos(π/3) = 1. 3. y = 2 sin(π/3) = √3. 4. Cartesian coordinates are (1, √3).

1. Use r = √(x² + y²) and θ = arctan(y/x). 2. r = √(1² + 1²) = √2. 3. θ = arctan(1/1) = π/4. 4. Polar coordinates are (√2, π/4).

1. Determine the period of the trigonometric function. 2. Period of sin(2θ) is π. 3. The period of the polar function is π.

1. Identify the limits of integration (0 to π). 2. Use the formula A = (1/2) ∫ [r(θ)]² dθ. 3. A = (1/2) ∫ (2cos(θ))² dθ from 0 to π. 4. Evaluate the integral to find the area.

1. Set the two equations equal to each other: sin(θ) = cos(θ). 2. Solve for θ: θ = π/4, 5π/4. 3. Find the corresponding r values. 4. The points of intersection are (√2/2, π/4) and (-√2/2, 5π/4).

1. Create a table of θ and r values for key angles (0, π/2, π, 3π/2, 2π). 2. Plot the points on the polar plane. 3. Connect the points to form the heart-shaped curve.

1. Replace θ with -θ in the equation. 2. If the equation remains unchanged, the curve is symmetric about the x-axis.

1. Find the maximum value of the trigonometric function. 2. The maximum value of sin(θ) is 1. 3. Therefore, the maximum value of r is 3.