The general form of polar coordinates :
(r, θ)
To convert polar coordinates to rectangular coordinates, substitute the given values of r and θ into the following.
x = r cos θ
y = r sin θ
Problems 1-5 : Convert the given polar coordinates to rectangular coordinates.
Problem 1 :
Solution :
The rectangular coordinates are
(√3, 1)
Problem 2 :
(-2, 45°)
Solution :
The rectangular coordinates are (-√2, -√2).
Problem 3 :
Solution :
The rectangular coordinates are
(-2√3, 2)
Problem 4 :
Solution :
The rectangular coordinates are
(3, -3√3)
Problem 5 :
(2, 480°)
Solution :
The rectangular coordinates are
(-1, √3)
The general form of polar coordinates :
(x, y)
To convert rectangular coordinates to polar coordinates, find the value of r using the formula given below.
To get the value of θ, first find the value of α we using the formula given below.
Based on the quadratnt in which we have the given rectangular coordinates, we can find the value of θ using the value of α as given below.
When α is in radians : Ist Quadrant : θ = α IInd Quadrant : θ = π - α IIIrd Quadrant : θ = π + α IVth Quadrant : θ = 2π - α On positive x-axis : θ = 0 On negative x-axis : θ = π On positive y-axis : On negative y-axis : |
When α is in degrees : Ist Quadrant : θ = α IInd Quadrant : θ = 180° - α IIIrd Quadrant : θ = 180° + α IVth Quadrant : θ = 360° - α On positive x-axis : θ = 0° On negative x-axis : θ = 180° On positive y-axis : θ = 90° On negative y-axis : θ = 270° |
Problems 6-12 : Convert the given rectangular coordinates to polar coordinates.
Problem 6 :
(1, √3)
Give the value of θ in radians.
Solution :
The value of r :
The value of θ :
The point (1, √3) is in Ist quadrant. Then, we have
θ = α
The polar coordinates are
Problem 7 :
(√3, -1)
Give the value of θ in radians.
Solution :
The value of r :
The value of θ :
The point (√3, -1) is in IVth Quadrant. Then, we have
The polar coordinates are
Problem 8 :
(-3, 3)
Give the value of θ in degrees.
Solution :
The value of r :
The value of θ :
The point (-3, 3) is in IInd Quadrant. Then, we have
θ = 180° - α
θ = 180° - 45°
θ = 135°
The polar coordinates are
(3√2, 135°)
Problem 9 :
(0, -2)
Give the value of θ in radians.
Solution :
The value of r :
The value of θ :
The point (0, -2) is on negative y-axis. Then, we have
The polar coordinates are
Problem 10 :
(-1, 0)
Give the value of θ in degrees.
Solution :
The value of r :
The value of θ :
The point (-1, 0) is on negative x-axis. Then, we have
θ = 180°
The polar coordinates are
(1, 180°)
Problem 11 :
(3, 0)
Give the value of θ in degrees.
Solution :
The value of r :
The value of θ :
The point (3, 0) is on positive x-axis. Then, we have
θ = 0°
The polar coordinates are
(3, 0°)
Problem 12 :
(-5, 12)
Give the value of θ in degrees.
Solution :
The value of r :
The value of θ :
The point (-5, 12) is in IInd Quadrant. Then, we have
θ = 180° - α
θ = 180° - 67.38°
θ = 112.62°
The polar coordinates are
(13, 112.62°)
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