Question

Help please :)) This is timed as well as mind boggling

Answer 1

Answer: Choice C

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Step-by-step explanation:

Recall that the form

`z = r*\left[\cos\left(\theta\right)+i*\sin\left(\theta\right)\right]`

can be abbreviated to

`z = r*\text{cis}(\theta)`

The "cis" stands for the first letters of "cosine i sine" in that order.

The original fairly messy quotient can be shortened down to

`\frac{90\text{cis}(\pi/4)}{2\text{cis}(\pi/12)}`

We have 90cis(pi/4) all over 2cis(pi/12) as one big fraction.

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Now we'll use this rule to divide two complex numbers

`z_1 = r_1*\text{cis}(\theta_1)\\\\z_2 = r_2*\text{cis}(\theta_2)\\\\(z_1)/(z_2) = (r_1)/(r_2)*\text{cis}(\theta_1-\theta_2)\\\\`

As you can see, we divide the r1 and r2 values to form the final coefficient out front. The theta angles are subtracted to form the new argument.

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Let's apply that idea to what your teacher gave you

`z_1 = 90*\text{cis}(\pi/4)\\\\z_2 = 2*\text{cis}(\pi/12)\\\\(z_1)/(z_2) = (90)/(2)*\text{cis}(\pi/4 - \pi/12)\\\\(z_1)/(z_2) = 45\text{cis}(3\pi/12 - \pi/12)\\\\(z_1)/(z_2) = 45\text{cis}(2\pi/12)\\\\(z_1)/(z_2) = 45\text{cis}(\pi/6)\\\\`

That last step then converts directly to the expression shown in choice C.