CONVERT COMPLEX NUMBERS FROM POLAR TO RECTANGULAR FORM

Example 1 :

Convert the given polar form to rectangular form

2(cos3Π/4  + isin3Π/4)

Solution :

By comparing the given polar form to the general equation of polar form r(cosθ + i sinθ), we get r = 2 and θ = 3Π/4. 

Rectangular form of a complex number is x + iy

x = rcosθ and y = rsinθ

Finding real part :

x = 2 cos 3Π/4 

3Π/4 = 135°

x = 2cos(90°+45°)

Since 135 lies in second quadratic, we have to put positive sign only for sin θ and its reciprocal cosec θ only. Here we have to put negative sign 

x = -2 sin 45° ==> -2 x 1/√2 ==> -√2

Hence the real part of the complex number is -√2. Now we have to find the imaginary part.

Finding imaginary part :

y = 2sin3Π/4 

y = 2cos(90°+45°)

y = 2sin45° ==> 2 x 1/√2 ==> √2

So the rectangular form of the complex number is 

-√2 + i√2

Example 2 :

Convert the given polar form to rectangular form

 2(cosΠ/3  + isinΠ/3)

Solution :

Rectangular form of a complex number is x + iy

x + iy =  2cosΠ/3  + i2sinΠ/3

x = rcosθ and y = rsinθ

r = 2 and  θ = Π/3 

Finding real part :

x = 2cosΠ/3  

Π/3 = 60°

Since 60° lies in the first quadrant, we have to put positive sign for all trigonometric ratios.

x = 2cos60° ==> 2 x 1/2 ==> 1

Hence the real part of the complex number is 1. Now we have to find the imaginary part.

Finding imaginary part :

y = 2 sin Π/3  

y = 2 sin 60° ==> 2 x (√3/2) ==> √3

So the rectangular form of the complex number is 

1 + i√3

Example 3 :

Convert the given polar form to rectangular form

2 (cos(-2Π/3)  + i sin(-2Π/3))

Solution :

Rectangular form of a complex number is x + iy

x + iy  =  2cos(-2Π/3)  + i2sin(-2Π/3)

x = rcosθ and y = rsinθ

r = 2 and  θ = -2Π/3 

Finding real part :

x = 2cos(-2Π/3)

2Π/3 = 120°

2cos(-2Π/3) = 2cos(-120°) ==> 2cos120°

2cos(90 + 30) ==> 2sin30° ==> 2 x (1/2) ==> 1

Hence the real part of the complex number is 1. Now we have to find the imaginary part.

Finding imaginary part :

y = 2sin(-2Π/3)

2sin(-2Π/3) = 2sin(-120°) ==> -2sin120°

-2sin(90 + 30) ==> -2cos30° ==> -2 x (√3/2) ==> -√3

So the rectangular form of the complex number is 

1 - i√3

Convert each equation from polar form to rectangular form :

Example 4 :

tan θ = 2

Solution :

tan θ = 2

tan θ = y/x -----(1)

In general, we know that x = r sin θ and y = r cos θ

By applying the value of tan θ in (1), we get

2 = y/x

2x = y

So, the rectangular form is y = 2x.

Example 5 :

r = 4 cos θ - 4 sin θ

Solution :

r = 4 cos θ - 4 sin θ

r = 4(cos θ - sin θ) ----(1)

we know that x = r sin θ and y = r cos θ

Solving for cos θ and sin θ,

x/r = sin θ and y/r = cos θ

Applying the values of cos θ and sin θ, we get 

r = 4(y/r - x/r)

r = 4(y - x)/r

r² = 4y - 4x

We know that r² = x² + y²

x² + y² = 4y - 4x

x² + y² + 4x - 4y = 0

x² + 4x + y² - 4y = 0

(x + 2)² - 4 + (y - 2)² - 4 = 0

(x + 2)² + (y - 2)² - 8 = 0

(x + 2)² + (y - 2)² = 8

Example 6 :

r = cot θ cosec θ

Solution :

r = cot θ cosec θ

Writing in terms of sin θ and cos θ

r = (cos θ/sin θ) (1/sin θ) -----(1)

we know that x = r sin θ and y = r cos θ

x/r = sin θ and y/r = cos θ

Applying the values of sin θ and cos θ in (1)

r = (y/r)/(x/r) (1/(x/r))

r = (y/r) ⋅ (r/x) (r/x)

r = (yr/x²)

x² = y

Example 7 :

r = -2 cos θ - 2 sin θ

Solution :

r = -2 cos θ - 2 sin θ -----(1)

we know that x = r sin θ and y = r cos θ

sin θ = x/r and cos cos θ = y/r

applying these values in (1), we get

r = -2 (y/r) + 2(x/r)

r = -2 (y + x)/r

r² = -2y - 2x

We know that r² = x² + y²

x² + y² = -2y - 2x

(x + 1)² + (y + 1)² = 2

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