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.