Trig Polar Stuff: Let z1 = -?(3) + i and z2 = 4 + 4i....

[quinnGoes]

Any of these problems that you could help me with would be great. the more the better =)

4. Let z1 = -?(3) + i and z2 = 4 + 4i

a. express z1,z2, and z1z2 in polar form
b. Show z1,z2, and z1,z2 in an argand diagram

5. Let z = 3 cis 150°

a. find z² in polar form and in rectangular form.
b. Show that z² in polar form agrees with z² in rectangular form.

6. a. Express z = 1 - i in polar form
b. Show z,z²,z^3, and z^4 in an Argand diagram.
c. Find z^10.

7. Write a paragraph giving a geometric interpretation of the n nth roots of a complex number.

8. Show that ³?(2) cis 130° is a cube root of ?(3) + i, and find the other two cube roots.

I'm in a higher math class and I'm trying to help a friend, but I don't remember this stuff. Thanks for any help!

 

[Deleted member 4993]

Any of these problems that you could help me with would be great. the more the better =)

4. Let z1 = -?(3) + i and z2 = 4 + 4i

a. express z1,z2, and z1z2 in polar form
b. Show z1,z2, and z1,z2 in an argand diagram

5. Let z = 3 cis 150°

a. find z² in polar form and in rectangular form.
b. Show that z² in polar form agrees with z² in rectangular form.

6. a. Express z = 1 - i in polar form
b. Show z,z²,z^3, and z^4 in an Argand diagram.
c. Find z^10.

7. Write a paragraph giving a geometric interpretation of the n nth roots of a complex number.

8. Show that ³?(2) cis 130° is a cube root of ?(3) + i, and find the other two cube roots.

I'm in a higher math class and I'm trying to help a friend, but I don't remember this stuff. Thanks for any help!

Please let your friend show us your work, indicating exactly where you (and your friend) are stuck, so that we know where to begin to help you.

 

[soroban]

Hello, quinnGoes!

Here's a start . . .

\(\displaystyle \text{5. Let }z \:=\:3\!\text{ cis}150^o\)

a. Find \(\displaystyle z^2\) in polar form and in rectangular form.

b. Show that \(\displaystyle z^2\) in polar form agrees with \(\displaystyle z^2\) in rectangular form.

\(\displaystyle \text{Polar form:}\)

\(\displaystyle \text{We have: }\;z \;=\;3\!\text{ cis}150^o\)

\(\displaystyle \text{Then: }\;z^2 \;=\;9\!\text{ cis }\!300^o\;=\;9\left(\cos300^o + i\sin300^o\right) \;=\;9\left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{2}i\right)\) .[1]


\(\displaystyle \text{Rectangular form:}\)

\(\displaystyle \text{We have: }\;z \;=\;3\!\text{ cis}150^o \;=\;3\left(\cos150^o + i\sin150^o\right) \;=\;3\left(\text{-}\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right)\)

\(\displaystyle \text{Then: }\;z^2 \;=\;3^2\left(\text{-}\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right)^2 \;=\;9\left(\tfrac{3}{4} - \tfrac{\sqrt{3}}{2}i - \tfrac{1}{4}\right) \;=\;9\left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{2}i\right)\) .[2]


And we see that: .[1] = [2].

 

[quinnGoes]

Hello, quinnGoes!

Here's a start . . .

\(\displaystyle \text{5. Let }z \:=\:3\!\text{ cis}150^o\)

a. Find \(\displaystyle z^2\) in polar form and in rectangular form.

b. Show that \(\displaystyle z^2\) in polar form agrees with \(\displaystyle z^2\) in rectangular form.

\(\displaystyle \text{Polar form:}\)

\(\displaystyle \text{We have: }\;z \;=\;3\!\text{ cis}150^o\)

\(\displaystyle \text{Then: }\;z^2 \;=\;9\!\text{ cis }\!300^o\;=\;9\left(\cos300^o + i\sin300^o\right) \;=\;9\left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{2}i\right)\) .[1]


\(\displaystyle \text{Rectangular form:}\)

\(\displaystyle \text{We have: }\;z \;=\;3\!\text{ cis}150^o \;=\;3\left(\cos150^o + i\sin150^o\right) \;=\;3\left(\text{-}\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right)\)

\(\displaystyle \text{Then: }\;z^2 \;=\;3^2\left(\text{-}\tfrac{\sqrt{3}}{2} + \tfrac{1}{2}i\right)^2 \;=\;9\left(\tfrac{3}{4} - \tfrac{\sqrt{3}}{2}i - \tfrac{1}{4}\right) \;=\;9\left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{2}i\right)\) .[2]


And we see that: .[1] = [2].

thanks man

 

[quinnGoes]

We managed to figure this out so don't worry about helping any more =)