Question

6 questions. let z1 = 2(cos(120°) + isin(120°)), z 2 = 4(cos(35°) + isin(35°)), z 3 = 5(cos(90°) + isin(90°)). Calculate the following keeping your answer in polar form

Answer 1

1. z1 + z2
2. z3 – z2 Answer:
3. z2z3 Answer:
4. z3 (straight line above)
5. z1/z3
6. z1/(z2z3) (use answer for 3.)

Answer 2

I have no Idea sorry here let @SolomonZelman help you

Answer 3

First convert z1 and z2 into x+iy form.

z1:
\[2[(\frac{ -1 }{ 2 }) + i \frac{ \sqrt{3} }{ 2 }]\]
z1= \[-1 + i \sqrt{3 }\]

Answer 4

I have to keep them in polar form

Answer 5

This way addition/ subtraction is easier. O.o

Answer 6

I know but the instructions say keep in polar form :(

Answer 7

See, for addition and subtraction, this is the easier way. You can then convert the answer you get back to polar form.

Answer 8

@primeralph

Answer 9

Just rewriting your question for easier viewing:
\[\begin{cases}
z_1=2\left(\cos120^\circ+i\sin120^\circ\right)\\
z_2=4\left(\cos35^\circ+i\sin35^\circ\right)\\
z_3=5\left(\cos90^\circ+i\sin90^\circ\right)
\end{cases}\]
Calculate the following:
\[\begin{align*}(1)&z_1+z_2\\
(2)&z_3-z_2\\
(3)&z_2z_3\\
(4)&\overline{z_3}\\
(5)&\frac{z_1}{z_3}\\
(6)&\frac{z_1}{z_2z_3}\end{align*}\]
I agree with @emcrazy14's method of converting to rectangular, then adding/subtracting, then converting back to polar for the first two.

(3), (5), and (6) are easily done in polar form, if you can recall the formula's I'd posted on one of your previous questions.

(4) is by far the easiest. If \(z=a+bi\), then what is its conjugate? If \(\large z=re^{i\theta}\), what is its conjugate?

Answer 10

@SithsAndGiggles yes I remember your formulas from my previous questions, so 3,5,6 I can do. I see a "math processing error" though on your posts, so I don't know what you typed. I just need help on 1,2,and 4

Answer 11

@phi can you help on 1,2,4?

Answer 12

I am not sure why it says "keeping your answer in polar form". they are currently in rectangular (a+bi) form.

I would add real to real and imaginary to imaginary.
then convert to polar form: find the magnitude of a+bi ( sqr(a^2 +b^2)
and angle atan(b/a)

Answer 13

For (4), if you're given a complex number \(z=a+bi\), then its conjugate is \(\overline{z}=a-bi\). In other words, given a complex number, the conjugate has the same real part and the negative imaginary part.

For (1) and (2), convert to rectangular:
\[\begin{cases}
z_1=2\left(\cos120^\circ+i\sin120^\circ\right)\\
z_2=4\left(\cos35^\circ+i\sin35^\circ\right)\\
z_3=5\left(\cos90^\circ+i\sin90^\circ\right)
\end{cases}~~\Rightarrow~~\begin{cases}
z_1=2\left(-\dfrac{1}{2}+\dfrac{\sqrt3}{2}i\right)=-1+\sqrt 3~i\\
z_2=\cdots\\
z_3=5(0+i)=5i
\end{cases}\]
For \(z_2\), are you sure the angle isn't 135, or 45?

Answer 14

@sithsandgiggles yes I'm sure. It is 35

Answer 15

Hmm well there's no easy way to compute it mentally, so use a calculator to determine the cosine and sine of 35 degrees.