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6 problems match “unit circle”. Clear

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1. Argument of a sum of exponentials ●●○○○ 20 ptsThe number $e^{7\pi i/60} + e^{17\pi i/60} + e^{27\pi i/60} + e^{37\pi i/60} + e^{47\pi i/60}$ is written in the form $re^{i\theta}$, where $r \ge 0$ and $0 \le \theta < 2\pi$. Find $\theta$ in degrees.
2. Real part of reciprocal on unit circle ●●○○○ 20 ptsLet $z$ be a complex number with $|z| = 1$ and $z \neq -1$. Show that $\operatorname{Re}\left(\frac{1}{z+1}\right)$ is a constant, and find its value.
3. Modulus of a symmetric complex ratio ●●○○○ 20 ptsLet $z_1, z_2, z_3$ be complex numbers with $|z_1|=|z_2|=|z_3|=1$, and assume $z_1+z_2+z_3\neq 0$. Find $\left|\frac{z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}}{z_{1}+z_{2}+z_{3}}\right|$.
4. Argument principle on a doubly traversed circle ●●○○○ 20 ptsLet $f(z)=z^3e^{z^2}$ for $z\in\mathbb{C}$, and let $\Gamma$ be the circle $z=e^{i\theta}$, where $0\leq\theta\leq4\pi$, so the contour goes around the unit circle twice counterclockwise. Find $\frac{1}{2\pi i}\oint_{\Gamma} \frac{f'(z)}{f(z)}\,dz$.
5. Sine plus cosine on a ray ●●○○○ 20 ptsGiven that the terminal side of angle $\theta$ lies on the ray $y=2x$ with $x\le 0$, find the value of $\sin\theta+\cos\theta$. Round to two decimal places.
6. Sums of squares on a unit circle ●●●○○ 50 ptsThree points $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$ lie on the unit circle $x^2+y^2=1$ and satisfy $x_1+x_2+x_3=0$ and $y_1+y_2+y_3=0$. Find $x_1^2+x_2^2+x_3^2$ and $y_1^2+y_2^2+y_3^2$.