Write each complex number in rectangular form?

I need help with this. :/

2(cos120°+i sin 120°)

and

4(cos 7π/4 + i sin 7π/4)

?

2*cos120 + i2*sin120.

X = hor = 2*cos120 = -1.
Y = VER = 2*sin120 = 1.732.

tanA = Y/X = 1.732/-1 = -1.732,
A = -60 deg = 300 deg CCW.

R = X/cosA = = -1/cos300 = -2 = Resultant.

R = -2 @ -60 deg = -2 @ 300 deg CCW.

4*cos7pi/4 + i4*sin7pi/4,

7pi/4 = 7*180/4 = 315 deg.
X = hor = 4*cos315 = 2.83.
Y = ver = 4*sin315 = -2.83.

tanA = Y/X = -2.83/2.83 = -1.
A = -45 deg = 315 deg CCW.

R = X/cosA = 2.83/cos315 = 4.0 = Resultant.

R = 4.0 @ -45 deg = 4.0 @ 315 deg CCW = Resultant.

To write a complex number in rectangular form, we can use Euler's formula, which states that for any angle θ:

e^(iθ) = cos(θ) + i sin(θ)

Let's start with the first complex number:

2(cos120° + i sin120°)

To convert it to rectangular form, we need to evaluate the cosine and sine of 120°.

cos(120°) = -0.5
sin(120°) = √3/2

Now, substitute these values back into the complex number:

2(-0.5 + i √3/2)

Simplifying:

-1 + i√3

Therefore, the first complex number, 2(cos120° + i sin120°), can be written as -1 + i√3 in rectangular form.

Now, let's move on to the second complex number:

4(cos 7π/4 + i sin 7π/4)

To convert it to rectangular form, we need to evaluate the cosine and sine of 7π/4.

cos(7π/4) = √2/2
sin(7π/4) = -√2/2

Now, substitute these values back into the complex number:

4(√2/2 + i (-√2/2))

Simplifying:

2√2 - 2i√2

Therefore, the second complex number, 4(cos 7π/4 + i sin 7π/4), can be written as 2√2 - 2i√2 in rectangular form.

I hope this helps! Let me know if you have any further questions.