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.
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
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.
To convert the polar form of a complex number into rectangular form, we need to use the formulas:
a = r * cos(theta)
b = r * sin(theta)
Given the polar form of the complex number z = 3√2 cis(7π/4), we can separate it into two parts - the magnitude (r) and the angle (theta).
Magnitude (r):
From the expression 3√2, we can see that the magnitude (r) is 3√2.
Angle (theta):
In the given polar form, cis(7π/4) represents the angle (theta).
To convert r and theta into a and b, we substitute the values into the formulas:
a = r * cos(theta)
b = r * sin(theta)
Using the values we determined:
a = (3√2) * cos(7π/4)
b = (3√2) * sin(7π/4)
Now, let's solve for a and b:
a = (3√2) * cos(7π/4)
= (3√2) * (-1/√2) (Using the unit circle values for cos and sin at 7π/4)
= -3
b = (3√2) * sin(7π/4)
= (3√2) * (1/√2) (Using the unit circle values for cos and sin at 7π/4)
= 3
Therefore, the rectangular form of the complex number z = 3√2 cis(7π/4) is a + bi, which is -3 + 3i.
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