Find the quotient of the following expression: 7(cos 5pi/6 +isin 5pi/6) / -2(cos pi/2 +isin pi/2)

I got -7/4+ 7 square root 3/4 i. Is this correct?

7(cos 5pi/6 +isin 5pi/6) / -2(cos pi/2 +isin pi/2)

= 7(-√3/2 + i/2) / (-2(0 + i))
= (7/2)( -√3 + i) / (-2i) * i/i
= (7/2)(-√3i + i^2)/(-2i^2)
= (7/2)(-1 - √3 i)/2
= (-7/4)(1 + √3 i)

your 7 square root 3/4 i part is ambiguous, it needs brackets to properly place the order of operation
I think you might have it correct

check my arithmetic, I should have written it out on paper for easier reading

To find the quotient of the given expression, you need to perform complex number division. Here's how you can do it step by step:

Step 1: Convert the complex numbers to their rectangular form.
The complex number in the numerator, 7(cos(5π/6) + i sin(5π/6)), can be represented in rectangular form as:
7(cos(5π/6) + i sin(5π/6)) = 7 * cos(5π/6) + i * (7 * sin(5π/6)) = 7 * (-√3/2) + i * (7/2) = (-7√3/2) + (7/2)i.

Similarly, the complex number in the denominator, -2(cos(π/2) + i sin(π/2)), can be represented in rectangular form as:
-2(cos(π/2) + i sin(π/2)) = -2 * cos(π/2) + i * (-2 * sin(π/2)) = -2 * (0) + i * (-2) = -2i.

Step 2: Divide the two complex numbers using the division operation in rectangular form.
((-7√3/2) + (7/2)i) / (-2i)

To divide complex numbers, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of -2i is 2i.

((-7√3/2) + (7/2)i) * (2i) / (-2i) * (2i)

Simplifying further:
((-7√3/2) + (7/2)i) * (2i) = (-7√3i + 7i^2) / (-2i)

Since i^2 is equal to -1:
(-7√3i + 7i^2) / (-2i) = (-7√3i - 7) / (-2i)

Step 3: Multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.

((-7√3i - 7) * (2i)) / (-2i * (2i))

Simplifying further:
((-7√3i - 7) * (2i)) = (-14i√3 - 14i) / (-4i^2)

Since i^2 is equal to -1:
(-14i√3 - 14i) / (-4i^2) = (-14i√3 - 14i) / (4)

Dividing both the numerator and denominator by 2:
((-14i√3 - 14i) / (4)) = (-7i√3 - 7i) / 2

Factoring out the common factor of i:
(-7i√3 - 7i) / 2 = -7i (√3 + 1) / 2

So, the quotient of the expression 7(cos(5π/6) + i sin(5π/6)) / -2(cos(π/2) + i sin(π/2)) is -7i(√3 + 1) / 2.

Therefore, your answer, -7/4 + (7√3/4)i, is incorrect. The correct answer is -7i(√3 + 1) / 2.