HOW TO CONVERT BETWEEN POLAR AND RECTANGULAR COORDINATES

Converting Polar Coordinates to Rectangular Coordinates

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)

Converting Rectangular Coordinates to Polar Coordinates

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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