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Convert Points Between Polar and Rectangular Coordinates quiz

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  • What are the formulas to convert a point (r, θ) in polar coordinates to rectangular coordinates (x, y)?
    x = r * cos(θ) and y = r * sin(θ).
  • How do you find the x-coordinate when converting from polar to rectangular coordinates?
    Multiply the radius r by the cosine of the angle θ: x = r * cos(θ).
  • How do you find the y-coordinate when converting from polar to rectangular coordinates?
    Multiply the radius r by the sine of the angle θ: y = r * sin(θ).
  • What is the rectangular coordinate of the polar point (5, π/3)?
    The rectangular coordinate is (5/2, (5√3)/2).
  • If a polar point has r = 0, what is its rectangular coordinate?
    It is always (0, 0), the origin.
  • What is the rectangular coordinate of the polar point (-3, π/6)?
    It is (-(3√3)/2, -3/2).
  • What is the general formula to convert rectangular coordinates (x, y) to polar coordinates (r, θ)?
    r = √(x² + y²) and θ = arctan(y/x), with adjustments for the correct quadrant.
  • How do you calculate the radius r when converting from rectangular to polar coordinates?
    Use the Pythagorean theorem: r = √(x² + y²).
  • How do you calculate the angle θ when converting from rectangular to polar coordinates?
    θ = arctan(y/x), but you must adjust θ based on the quadrant where (x, y) is located.
  • What is the polar coordinate of the rectangular point (3, 4)?
    It is (5, arctan(4/3)), or approximately (5, 53°).
  • What adjustment must you make to θ if your rectangular point is in quadrant II or III?
    Add π to the angle θ found from arctan(y/x) to place it in the correct quadrant.
  • What is the polar coordinate of the rectangular point (-4, 0)?
    It is (4, π).
  • What is the polar coordinate of the rectangular point (-1, √3)?
    It is (2, 2π/3).
  • Why is it important to consider the quadrant when converting from rectangular to polar coordinates?
    Because the arctan function only gives angles in quadrants I and IV, so you must adjust θ to match the actual location of the point.
  • What is the relationship between the unit circle and converting polar to rectangular coordinates?
    The formulas x = r cos(θ) and y = r sin(θ) are based on the unit circle, but r can be any value, not just 1.