Could someone show me how to solve these two problems-I have 10 of each kind to do this weekend and I'm really lost

1. Find product of 8(cos pi/2 + i sin pi/2) * 1/2(cos pi/6 + i sin pi/6)

2. Same problem but find quotient instead of product
Please someone show me these two so I can do my other problems from these examples
Thankyou Thank you

1. That can be written

8*e^(i*pi/2) * (1/2)e^(i pi/6)
= 4 e^(2i*pi/3)
= 4 [cos(2*pi/3) + i sin (2*pi/3)]
= -2 + 2(sqrt3)i

2. The quotient is
16 e^(i*pi/3)
= 16[cos(pi/3) + i sin(pi/3)]
= 8 + 8 sqrt3 i

Thank you very much for showing me the steps but they don't match my answers that are available: Could you possibly further explain-I'm messed up right now on this concept-

the answers for Number 1 are
4cis(2pi/3)
4cis(pi/3)
4cis(pi/8)
4cis(pi/4)
2. answers for number two are:
16cis(pi/3)16cis(2pi/3)
4cis(2pi/3)
4cis(pi/3)

cis Theta = cos Theta + i sin Theta

x = r cos Theta
y = r sin Theta

so for example
4 cis 2 pi/3 = 4 cos 2 pi/3 + 4 i sin 2pi/3
but cos 2p/i/3 = -.5 and sin 2pi/3 = sqrt 3/2
so
x = 4 (-.5) = -2
y = 4 (.5/ sqrt 3) = 2 sqrt3
-2 + 2 i sqrt 3
Your answer is the same as that of WLS

Of course! I'll be glad to help you solve these problems step by step.

To solve the first problem, which involves finding the product of two complex numbers, you can follow these steps:

Step 1: Rewrite both complex numbers in rectangular form.

The first complex number, 8(cos π/2 + i sin π/2), can be expressed as 8 * cis(π/2) in rectangular form.

The second complex number, 1/2(cos π/6 + i sin π/6), can be expressed as 1/2 * cis(π/6) in rectangular form.

Step 2: Multiply the two complex numbers.

To multiply complex numbers, you can multiply their magnitudes and add their arguments.

The magnitudes of the two complex numbers are 8 and 1/2, so their product gives you 8 * 1/2, which simplifies to 4.

The arguments of the two complex numbers are π/2 and π/6. Adding them together gives you π/2 + π/6 = 4π/6 + π/6 = 5π/6.

Step 3: Convert the result back to polar form.

The product of the two complex numbers, 4 * cis(5π/6), can be expressed as 4(cos 5π/6 + i sin 5π/6) in polar form.

Therefore, the product of 8(cos π/2 + i sin π/2) and 1/2(cos π/6 + i sin π/6) is 4(cos 5π/6 + i sin 5π/6).

Now, let's move on to the second problem and find the quotient (division) of the two complex numbers.

Following similar steps:

Step 1: Rewrite both complex numbers in rectangular form.

The first complex number, 8(cos π/2 + i sin π/2), can be expressed as 8 * cis(π/2) in rectangular form.

The second complex number, 1/2(cos π/6 + i sin π/6), can be expressed as 1/2 * cis(π/6) in rectangular form.

Step 2: Divide the two complex numbers.

To divide complex numbers, you can divide their magnitudes and subtract their arguments.

The magnitudes of the two complex numbers are 8 and 1/2, so their quotient gives you 8 / 1/2, which simplifies to 16.

The arguments of the two complex numbers are π/2 and π/6. Subtracting them gives you π/2 - π/6 = 3π/6 - π/6 = 2π/6.

Step 3: Convert the result back to polar form.

The quotient of the two complex numbers, 16 * cis(2π/6), can be expressed as 16(cos 2π/6 + i sin 2π/6) in polar form.

Therefore, the quotient of 8(cos π/2 + i sin π/2) and 1/2(cos π/6 + i sin π/6) is 16(cos 2π/6 + i sin 2π/6).

Now you can use these steps to solve the rest of your problems! Let me know if there's anything else I can assist you with.