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Convert Points Between Polar and Rectangular Coordinates quiz #1 Flashcards

Convert Points Between Polar and Rectangular Coordinates quiz #1
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  • What are the rectangular coordinates (x, y) of a point given its polar coordinates (r, θ)?
    The rectangular coordinates are x = r * cos(θ) and y = r * sin(θ).
  • How do you convert a point from cylindrical coordinates (r, θ, z) to spherical coordinates (ρ, φ, θ)?
    To convert from cylindrical coordinates (r, θ, z) to spherical coordinates (ρ, φ, θ): ρ = √(r² + z²), φ = arccos(z / ρ), and θ remains the same.
  • What is the rectangular coordinate of a polar point with r = 0 and any angle θ?
    The rectangular coordinate will always be (0, 0). This is because multiplying r = 0 by any trigonometric value results in zero for both x and y.
  • How do you determine the sign of r when plotting a polar point?
    A negative r value means you plot the point in the direction opposite to the angle θ. This places the point in the reflected position across the origin.
  • What is the first step when converting a rectangular point to polar coordinates?
    The first step is to plot the point on the rectangular coordinate system. This helps you identify its quadrant and location.
  • Why must you adjust θ when converting rectangular coordinates to polar coordinates for points in quadrant 2?
    You must add π to the angle found using inverse tangent to ensure θ is in the correct quadrant. This adjustment places the angle in quadrant 2 where the point is located.
  • What is the polar coordinate of the rectangular point (-4, 0)?
    The polar coordinate is (4, π). The r value is 4 and θ is π because the point lies on the negative x-axis.
  • How do you find θ for a rectangular point not on the axes?
    Calculate θ using the inverse tangent of y/x, then adjust based on the quadrant where the point is located. This ensures the angle matches the actual position of the point.
  • What is the rectangular coordinate of the polar point (-3, π/6)?
    The rectangular coordinate is (-3√3/2, -3/2). This is found by multiplying r by the cosine and sine of θ.
  • What is the polar coordinate of the rectangular point (-1, √3)?
    The polar coordinate is (2, 2π/3). The r value is 2 and θ is 2π/3 after adjusting for the correct quadrant.