### Instruction: The roots of
\[z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\]are $\text{cis } \theta_1$, $\text{cis } \theta_2$, $\dots$, $\text{cis } \theta_7$, where $0^\circ \le \theta_k < 360^\circ$ for all $1 \le k \le 7$.  Find $\theta_1 + \theta_2 + \dots + \theta_7$.  Give your answer in degrees.### Response: I recognize that the right-hand side of the equation is a complex number in polar form, with modulus 1 and argument 225 degrees.
So I can rewrite the equation as \[z^7 = \text{cis } 225^\circ.\]
Now, I recall that De Moivre's theorem says that if $z = r \text{cis } \theta$, then $z^n = r^n \text{cis } (n\theta)$.
So, to find the seventh roots of $\text{cis } 225^\circ$, I need to divide the argument by 7 and add multiples of $\frac{360}{7}$ degrees to get all the possible values.
That is, \[z = \text{cis } \frac{225}{7}^\circ, \text{cis } \left(\frac{225}{7} + \frac{360}{7}\right)^\circ, \dots, \text{cis } \left(\frac{225}{7} + \frac{6 \cdot 360}{7}\right)^\circ.\]
These are the values of $\text{cis } \theta_1$, $\text{cis } \theta_2$, $\dots$, $\text{cis } \theta_7$.
To find their sum, I need to add up the arguments of these complex numbers.
So, \[\theta_1 + \theta_2 + \dots + \theta_7 = \frac{225}{7} + \left(\frac{225}{7} + \frac{360}{7}\right) + \dots + \left(\frac{225}{7} + \frac{6 \cdot 360}{7}\right).\]
This simplifies to \[\theta_1 + \theta_2 + \dots + \theta_7 = 7 \cdot \frac{225}{7} + \frac{360}{7} \cdot \frac{7 \cdot 6}{2}.\]
This further simplifies to \[\theta_1 + \theta_2 + \dots + \theta_7 = 225 + 1080.\]
So, the final answer is \[\theta_1 + \theta_2 + \dots + \theta_7 = 1305^\circ.\]