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1. Find x+y from a trigonometric system ●●○○○ 20 ptsGiven that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$. Round to two decimal places.
2. Sine of a double angle from a shifted sine ●●○○○ 20 ptsGiven $\alpha \in (0, \frac{\pi}{2})$ and $\sin (2\alpha - \frac{\pi}{3}) = \frac{\sqrt{3}}{3}$, find $\sin 2\alpha$. Round to four decimal places.
3. Maximum of x+y under a trig constraint ●●○○○ 20 ptsFind the maximum value of $x+y$ if $(2\sin x-1)(2\cos y-\sqrt{3})=0$, $x\in[0,3\pi/2]$, $y\in[\pi,2\pi]$. Round to two decimal places.
4. Sum of x and y in a system ●●●○○ 50 ptsGiven that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$. Give your answer as a decimal rounded to two decimal places.
5. Unit complex numbers with a Re-Im condition ●●●○○ 50 ptsHow many complex numbers $z$ satisfy the following two conditions: $|z|=1$ and $\operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right)$?
6. Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers $x$ for which all six functions are defined, find the minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|$. Round your answer to two decimal places.
7. Average value of a trigonometric function ●●●○○ 50 ptsConsider the function $f(x) = 4\sin x - 2\sin 2x$ on the interval $[0, \pi]$. (a) Find the average value $f_{\text{ave}}$ of $f$ on this interval. (b) Find the two values $c_1 < c_2$ in $[0, \pi]$ such that $f(c_1) = f(c_2) = f_{\text{ave}}$. Round your answers to three decimal places.