write the following in the polar form

Question

Write the following in the polar form r^eiθ, -pie<0≤pie
a)πi 
r=
θ=
b)-2√2- 5i
r=
θ=
c) (1-i)(-√(3)+i))
r=
θ=
d)(√(2)-5i))2
r=
θ=
e)(-5+√(2i))/(5+3i)
r=
θ=
f) -√7(1+i)/ √(3) +i
r=
θ=

Answer 1

Dalia, yesterday you posed the same question here:
http://www.wyzant.com/resources/answers/27705/

Here’s my answer, again:
[I would like other tutors to critique it for errors, though.]

Write the following in the polar form re^(iθ), -pi < 0 ≤ pi

a) pi i = re^(iθ), where:
r = pi
θ = pi/2

b) -2√2 - 5i = re^(iθ), where:
r = √(8+25) = √(33)
θ = -pi + arctan((5√(2))/4)

c) (1-i)(-√(3)+i)) = (√(2) e^(-i pi/4)) (2 e^(i 5pi/6))
= 2√(2) e^(i 7pi/12)
r = 2√(2)
θ = 7pi/12

d)(√(2)-5i))^2 = ( √(27) e^(-i arctan( (5√(2))/2) ) ) )^2
= 27 e^(-2i arctan( (5√(2))/2) ) )
r = 27
θ = -2 arctan( (5√(2))/2) )

e)(-5+√(2i))/(5+3i)

2i = 2e^(i pi/2)
√(2i) = √(2) e^(i pi/4) = 1 + i

-5+√(2i) = -4 + i = √(17) e^(i (pi + arctan(-1/4)))
5+3i = √(34) e^(i arctan(3/5))

(-5+√(2i))/(5+3i) = ( √(17) e^(i (pi + arctan(-1/4))) )/( √(34) e^(i arctan(3/5)) )
= (√(2))/2 e^(i ( (pi + arctan(-1/4))) - (arctan(3/5)) ) )

r = (√(2))/2
θ = pi + arctan(-1/4) - arctan(3/5) = pi + arctan(7/23)

f) -√7(1+i)/(√(3)+i)

- √7(1+i) = - √(7) √(2) e^(i pi/4)
√(3)+i = 2 e^(i arctan( (√(3))/3 ) )
-√7(1+i)/(√(3)+i) = - √(14) e^(i pi/4) / ( 2 e^(i arctan( (√(3))/3 ) ) )

r = (√(14))/2
θ = pi/4 - arctan( (√(3))/3 )

Answer 2

Each initial statement is a complex number.

 

Simplify the number so it has the form x + yi

 

 

You can think of a complex number as a point on an x-y (horizontal-vertical) plane, with the real part along the horizontal and the imaginary part along the vertical.

 

Just like back in the day, the x (real) and y (imaginary) components (Cartesian or rectangular form) can be translated to a radial distance from the origin, "r", and an angle measured positive counter-clockwise from the positive horizontal axis to the negative horizontal axis and measured negative clockwise from positive horizontal axis to the negative horizontal axis, "θ", (polar form), using trigonometry. The angle is measured in radians, and 2π radians is a full rotation.

 

I find it helps to think of a few "extreme" examples to help sort this out.

 

If a point lies at x = 10 and y = 0, then the radial distance "r" from the origin is 10, and the line segment from the origin to the point is along the positive horizontal axis is 0. Therefore, x,y (10,0) -> r,θ (10,0)

 

If a point lies at x = 0 and y = 10, then r = 10, again. This time the line segment from the origin to the point is along the positive vertical axis. This is 90 degrees counter-clockwise around from the positive horizontal axis, a quarter rotation, so θ = 90 degrees = (2π)/4 radians = π/2 radians.

 

If a point lies at x = -10 and y = 0, then r = 10, again, because it's the length of the line segment, with no direction therefore never written as a negative number. The line segment lies along the negative horizontal axis. This is 180 degrees from the positive horizontal axis, a half rotation positive or negative, so θ = 180 degrees = (2π)/2 = π radians or -180 degrees, -(2π)/2 = -π radians.

 

One last extreme example. If a point lies at x = 0 and y = -10, then r = 10, again. This time the line segment from the origin to the point is along the negative vertical axis. This is 90 degrees clockwise around from the positive horizontal axis, a negative quarter rotation, so θ = -90 degrees = -(2π)*1/2 radians = -π radians.

 

So, these are the easy ones.

 

If your complex number is a positive real number with no complex component (a + 0i), then r = a and θ = 0.

If a negative real number with no complex component (-a + 0i), then r = a and θ = π.

If no real component and positive complex component (0 + ai), then r = 0 and θ = π/2.

If no real component and negative complex component (0 - ai), then r = 0 and θ = -π.

 

 

For all others, use the pythagorean theorem for radial distance, r = sqrt(x^2 + y^2), and use tangent for angle, θ = arctan2(y/x), making sure to use the correct positive and negative signs to determine the value, and using radians.

 

Simple examples are for positive and negative values of 1.

 

x = +1, y = +1

r = sqrt((+1)^2 + (+1)^2) =  sqrt(1 + 1)  = sqrt(2) -> r = sqrt(2)

θ = arctan((+1)/(+1)) = π/4

 

x = -1, y = +1
r = sqrt((-1)^2 + (+1)^2) = sqrt(1 + 1) = sqrt(2) -> r = sqrt(2)
θ = arctan((+1)/(-1)) = π*3/4

 

x = -1, y = -1
r = sqrt((-1)^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2) -> r = sqrt(2)
θ = arctan((-1)/(-1)) = -π*3/4

 

x = +1, y = -1
r = sqrt((+1)^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2) -> r = sqrt(2)
θ = arctan((-1)/(+1)) = -π/4

 

 

For more on arctan2, see http://en.wikipedia.org/wiki/Atan2