Apply De Moivre's theorem to calculate the following and leave your answer in polar form with a positive argument: $$ \frac{(\sqrt{3}+i)^{2}}{\left(2 \text { cis } 35^{\circ}\right)^{3}} $$

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To solve the expression

$$
\frac{(\sqrt{3}+i)^{2}}{\left(2 \text{ cis } 35^{\circ}\right)^{3}},
$$

we will first convert $$\sqrt{3} + i$$ into polar form, then apply De Moivre's theorem to both the numerator and the denominator.

### Step 1: Convert $$\sqrt{3} + i$$ to polar form

1. **Magnitude**:
   $$
   r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2.
   $$

2. **Argument**:
   $$
   \theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \text{ (or } 30^\circ\text{)}.
   $$

Thus, we can express $$\sqrt{3} + i$$ in polar form as:
$$
\sqrt{3} + i = 2 \text{ cis } 30^\circ.
$$

### Step 2: Calculate $$(\sqrt{3} + i)^2$$

Using De Moivre's theorem:
$$
(\sqrt{3} + i)^2 = (2 \text{ cis } 30^\circ)^2 = 2^2 \text{ cis } (2 \cdot 30^\circ) = 4 \text{ cis } 60^\circ.
$$

### Step 3: Calculate $$\left(2 \text{ cis } 35^\circ\right)^3$$

Again, using De Moivre's theorem:
$$
\left(2 \text{ cis } 35^\circ\right)^3 = 2^3 \text{ cis } (3 \cdot 35^\circ) = 8 \text{ cis } 105^\circ.
$$

### Step 4: Combine the results

Now we substitute back into the original expression:
$$
\frac{(\sqrt{3}+i)^{2}}{\left(2 \text{ cis } 35^{\circ}\right)^{3}} = \frac{4 \text{ cis } 60^\circ}{8 \text{ cis } 105^\circ}.
$$

### Step 5: Simplify the fraction

1. **Magnitude**:
   $$
   \frac{4}{8} = \frac{1}{2}.
   $$

2. **Argument**:
   $$
   60^\circ - 105^\circ = -45^\circ.
   $$

Thus, we have:
$$
\frac{4 \text{ cis } 60^\circ}{8 \text{ cis } 105^\circ} = \frac{1}{2} \text{ cis } (-45^\circ).
$$

### Step 6: Convert the argument to a positive angle

To express the argument in a positive form, we can add $$360^\circ$$:
$$
-45^\circ + 360^\circ = 315^\circ.
$$

### Final Answer

Thus, the final result in polar form with a positive argument is:
$$
\frac{1}{2} \text{ cis } 315^\circ.
$$
$$ \frac{1}{2} \text{ cis } 315^\circ. $$

Apply De Moivre’s theorem to calculate the following and leave your answer in polar
form with a positive argument:

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