CONVERTING POLAR COORDINATES TO RECTANGULAR COORDINATES

Let P be the point have the polar coordinates (r, θ) and its rectangular coordinates will be (x, y).

Then,

x = rcosθ and y = rsinθ

r² = x² + y²

tanθ = y/x

Example 1 :

Convert the given polar coordinates to rectangular coordinates.

(a) (-4, 2π/3) (b) (√3, π/6)

Solution :

From the point (-4, 2π/3), r is -4 and θ is 2π/3

x = r cos θ

x = -4 cos 2π/3

x = -4(-1/2)

x = 2

y = r sin θ

y = -4 sin 2π/3

y = -4(√3/2)

y = -2√3

So, the required rectangular co ordinate is (2, -2√3).

(b) (√3, π/6)

From the point (√3, π/6), r is √3 and θ is π/6

x = r cos θ

x = √3 cos π/6

x = √3 (√3/2)

x = 3/2

y = r sin θ

y = √3 sin π/6

y = √3 (1/2)

y = √3/2

So, the required rectangular co ordinate is (3/2, √3/2).

Example 2 :

Convert the given polar coordinates to rectangular coordinates.

(a) (2, π/4) (b) (-3, 5π/6)

Solution :

(a) (2, π/4)

From the point (2, π/4), r is 2 and θ is π/4

x = r cos θ

x = 2 cos π/4

x = 2(1/√2)

x = 2/√2

y = r sin θ

y = 2 sin π/4

y = 2(1/√2)

y = 2/√2

So, the required rectangular co ordinate is (2/√2, 2/√2).

(b) (-3, 5π/6)

From the point (-3, 5π/6), r is -3 and θ is 5π/6

x = r cos θ

x = -3 cos 5π/6

x = -3(√3/2)

x = -3√3/2

y = r sin θ

y = -3 cos 5π/6

y = -3(1/2)

y = -3/2

So, the required rectangular co ordinate is

(-3√3/2, -3/2).

From the point (5, 10π/3), r is 5 and θ is 10π/3

x = r cos θ

x = 5 cos 10π/3

x = 5(-√3/2)

x = -5√3/2

y = r sin θ

y = 5 cos 10π/3

y = 5(-1/2)

y = -5/2

So, the required rectangular co ordinate is

(-5√3/2, -5/2).

(d) (47, 17π/2)

From the point (47, 17π/2), r is 47 and θ is 17π/2

x = r cos θ

x = 47 cos 17π/2

x = 47(0)

x = 0

y = r sin θ

x = 47 sin 17π/2

x = 47(1)

x = 47

So, the required rectangular co ordinate is (0, 47).

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CONVERTING POLAR COORDINATES TO RECTANGULAR COORDINATES

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