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Polar Coordinates quiz

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  • What are the two components of a point in polar coordinates?
    A point in polar coordinates is given by (r, θ), where r is the distance from the pole and θ is the angle from the polar axis.
  • How do you plot a point with a negative r value in polar coordinates?
    Plot the angle θ first, then move |r| units in the direction opposite to θ.
  • What is the relationship between coterminal angles and polar coordinates?
    Coterminal angles represent the same direction, so adding or subtracting multiples of 2π to θ gives different ordered pairs for the same point.
  • How do you find another polar coordinate for the same point if you change r to negative?
    Change r to -r and add π to θ, then you can also add or subtract multiples of 2π to θ.
  • What are the formulas to convert from polar to rectangular coordinates?
    x = r cos(θ) and y = r sin(θ).
  • How do you convert the rectangular point (x, y) to polar coordinates?
    r = sqrt(x^2 + y^2) and θ = arctan(y/x), adjusting θ based on the quadrant.
  • What is the rectangular coordinate of the polar point (0, θ)?
    It is always (0, 0), regardless of θ.
  • What is the polar form of the equation x^2 + y^2 = 25?
    The polar form is r^2 = 25.
  • How do you convert the rectangular equation y = 5 to polar form?
    Replace y with r sin(θ) to get r sin(θ) = 5, or r = 5 csc(θ).
  • What is the rectangular form of the polar equation r = 4?
    It is x^2 + y^2 = 16, which is a circle of radius 4.
  • How do you recognize the equation of a cardioid in polar coordinates?
    A cardioid has the form r = a ± a cos(θ) or r = a ± a sin(θ), where the coefficients are equal.
  • What is the general form of a rose curve in polar coordinates and how do you determine the number of petals?
    The form is r = a cos(nθ) or r = a sin(nθ); if n is even, there are 2n petals, if n is odd, there are n petals.
  • How do you identify a lemniscate from its polar equation?
    A lemniscate has the form r^2 = a^2 cos(2θ) or r^2 = a^2 sin(2θ), and is the only common polar shape with r squared.
  • What symmetry does the graph of r = 1 + cos(θ) have?
    It is symmetric about the polar axis (the line θ = 0).
  • When graphing a limacon r = a ± b sin(θ), how do you know if it has a dimple or an inner loop?
    If a > b, the limacon has a dimple; if a < b, it has an inner loop.