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What is the general form of a polar function?

$r = f(\theta)$

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What is the general form of a polar function?

$r = f(\theta)$

What is the polar equation of a circle passing through the origin with diameter 'a' along the x-axis?

$r = a \cos(\theta)$

What is the polar equation of a circle passing through the origin with diameter 'a' along the y-axis?

$r = a \sin(\theta)$

What is the equation for a spiral in polar coordinates?

$r = a\theta$ , where 'a' is a constant.

What is the general form of a cardioid?

$r = a(1 ± \cos \theta)$ or $r = a(1 ± \sin \theta)$

What is the general form of a lemniscate?

$r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$

How do you convert from polar coordinates (r, θ) to Cartesian coordinates (x, y)?

$x = r \cos(\theta)$ , $y = r \sin(\theta)$

How do you convert from Cartesian coordinates (x, y) to polar coordinates (r, θ)?

$r = \sqrt{x^2 + y^2}$ , $\theta = \arctan(\frac{y}{x})$

What is the formula for finding the area enclosed by a polar curve r = f(θ) from θ = α to θ = β?

$A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 d\theta$

What is the formula for arc length of a polar curve r = f(θ) from θ = α to θ = β?

$L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta$

How do you graph r = 2cos(θ)?

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

How do you determine the symmetry of r = f(θ)?

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.

How do you convert (2, π/3) from polar to Cartesian coordinates?

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).

How do you convert (1, 1) from Cartesian to polar coordinates?

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).

How do you find the period of r = sin(2θ)?

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

How do you find the area enclosed by the polar curve r = 2cos(θ)?

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.

How do you find the points of intersection between r = sin(θ) and r = cos(θ)?

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).

How do you graph a cardioid r = 1 + cos(θ)?

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.

How do you determine if a polar curve is symmetric about the x-axis?

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

How do you find the maximum value of r for the polar function r = 3sin(θ)?

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.

What does the graph of r = θ represent?

A spiral that extends outward from the origin as θ increases.

What shape does r = sin(θ) create?

A circle that touches the origin with a diameter along the y-axis.

What does the graph of r = 2cos(θ) represent?

A circle with diameter 2 along the x-axis, passing through the origin.

How can you identify symmetry about the x-axis from a polar graph?

The graph is symmetric about the x-axis if the portion above the x-axis is a mirror image of the portion below it.

How can you identify symmetry about the y-axis from a polar graph?

The graph is symmetric about the y-axis if the portion to the right of the y-axis is a mirror image of the portion to the left of it.

What does a cardioid graph look like?

A heart-shaped curve with a cusp at the origin.

What does a lemniscate graph look like?

A figure-eight-shaped curve centered at the origin.

How does the coefficient 'a' in r = a sin(θ) affect the circle's graph?

It determines the diameter of the circle. A larger 'a' means a larger diameter.

What does the graph of r = a (where a is a constant) represent?

A circle centered at the origin with radius 'a'.

What happens to the graph of r = f(θ) when 'r' values are negative?

The points are reflected through the origin, changing the direction of the radius.

All Flashcards

What is the general form of a polar function?

$r = f(\theta)$

What is the polar equation of a circle passing through the origin with diameter 'a' along the x-axis?

$r = a \cos(\theta)$

What is the polar equation of a circle passing through the origin with diameter 'a' along the y-axis?

$r = a \sin(\theta)$

What is the equation for a spiral in polar coordinates?

$r = a\theta$ , where 'a' is a constant.

What is the general form of a cardioid?

$r = a(1 ± \cos \theta)$ or $r = a(1 ± \sin \theta)$

What is the general form of a lemniscate?

$r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta)$

How do you convert from polar coordinates (r, θ) to Cartesian coordinates (x, y)?

$x = r \cos(\theta)$ , $y = r \sin(\theta)$

How do you convert from Cartesian coordinates (x, y) to polar coordinates (r, θ)?

$r = \sqrt{x^2 + y^2}$ , $\theta = \arctan(\frac{y}{x})$

What is the formula for finding the area enclosed by a polar curve r = f(θ) from θ = α to θ = β?

$A = \frac{1}{2} \int_{\alpha}^{\beta} [f(\theta)]^2 d\theta$

What is the formula for arc length of a polar curve r = f(θ) from θ = α to θ = β?

$L = \int_{\alpha}^{\beta} \sqrt{r^2 + (\frac{dr}{d\theta})^2} d\theta$

How do you graph r = 2cos(θ)?

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

How do you determine the symmetry of r = f(θ)?

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.

How do you convert (2, π/3) from polar to Cartesian coordinates?

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).

How do you convert (1, 1) from Cartesian to polar coordinates?

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).

How do you find the period of r = sin(2θ)?

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

How do you find the area enclosed by the polar curve r = 2cos(θ)?

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.

How do you find the points of intersection between r = sin(θ) and r = cos(θ)?

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).

How do you graph a cardioid r = 1 + cos(θ)?

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.

How do you determine if a polar curve is symmetric about the x-axis?

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

How do you find the maximum value of r for the polar function r = 3sin(θ)?

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.

What does the graph of r = θ represent?

A spiral that extends outward from the origin as θ increases.

What shape does r = sin(θ) create?

A circle that touches the origin with a diameter along the y-axis.

What does the graph of r = 2cos(θ) represent?

A circle with diameter 2 along the x-axis, passing through the origin.

How can you identify symmetry about the x-axis from a polar graph?

The graph is symmetric about the x-axis if the portion above the x-axis is a mirror image of the portion below it.

How can you identify symmetry about the y-axis from a polar graph?

The graph is symmetric about the y-axis if the portion to the right of the y-axis is a mirror image of the portion to the left of it.

What does a cardioid graph look like?

A heart-shaped curve with a cusp at the origin.

What does a lemniscate graph look like?

A figure-eight-shaped curve centered at the origin.

How does the coefficient 'a' in r = a sin(θ) affect the circle's graph?

It determines the diameter of the circle. A larger 'a' means a larger diameter.

What does the graph of r = a (where a is a constant) represent?

A circle centered at the origin with radius 'a'.

What happens to the graph of r = f(θ) when 'r' values are negative?

The points are reflected through the origin, changing the direction of the radius.