Question

Convert the polar form of the complex number into rectangular form.

z = 3 sqrt 2cis 7pi/4


Show all your work. Show how you separately convert r and theta into a and b.

Answer 1

AAAaannndd the bot gets it wrong yet again!
7π/4 is in QIV, so cos > 0 and sin < 0
You got your formulas right, but the final calculations are totally messed up.

3√2 cos 7π/4 = (3√2)(1/√2) = 3
3√2 sin 7π/4 = (3√2)(-1/√2) = -3
so 3√2 cis 7π/4 = 3 - 3i

Answer 2

To convert the polar form of a complex number to rectangular form, we use the formulas:

a = r cos(theta)
b = r sin(theta)

where r is the modulus (or absolute value) of the complex number, and theta is the argument (or angle) of the complex number.

In this case, we have:

r = 3 sqrt 2
theta = 7pi/4

To convert r and theta into a and b, we plug them into the formulas:

a = r cos(theta)
a = (3 sqrt 2) cos(7pi/4)
a = (3 sqrt 2) (-sqrt 2/2) (using the values of cos and sin for 7pi/4)
a = -9/2

b = r sin(theta)
b = (3 sqrt 2) sin(7pi/4)
b = (3 sqrt 2) (-sqrt 2/2) (using the values of cos and sin for 7pi/4)
b = -9/2

Therefore, the rectangular form of the complex number z is:

z = -9/2 - 9/2i

Answer 3

I apologize for the mistake in my previous response. You are correct that 7π/4 lies in the fourth quadrant, where cosine is positive and sine is negative. Therefore, the correct calculations are:

a = r cos(theta)
a = (3√2) cos(7π/4)
a = (3√2)(1/√2)
a = 3

b = r sin(theta)
b = (3√2) sin(7π/4)
b = (3√2)(-1/√2)
b = -3

Hence, z = 3 - 3i is the correct rectangular form of the complex number. Thank you for bringing this to my attention.