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1. Double-angle sine from given sine and cosine ●○○○○ 10 ptsIf $\theta$ is an angle in the first quadrant with $\sin(\theta) = \frac{3}{5}$ and $\cos(\theta) = \frac{4}{5}$, find $\sin(2\theta)$. Give your answer as a decimal.
2. Difference of cosines at 36 and 72 degrees ●●○○○ 20 ptsCompute $\cos 36^\circ - \cos 72^\circ$. Give your answer as a decimal.
3. Simplify expression with imaginary cosine and sine ●●●○○ 50 ptsSimplify $2\cos^2(i\log 2009)+i\sin(i\log 4036081)$, where $\log$ is the natural logarithm. Give your answer as a decimal rounded to eight decimal places.
4. 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 $y_1^2+y_2^2+y_3^2$.
5. RMS voltage of household current ●●●○○ 50 ptsHousehold electricity has voltage $E(t)=155\sin(120\pi t)$ volts, where $t$ is time in seconds, and one cycle lasts $\frac{1}{60}$ second. The RMS voltage is the square root of the average value of $[E(t)]^2$ over one cycle. Find the RMS voltage of household current in volts. Round to one decimal place.
6. Sine of a shifted double angle ●●●○○ 50 ptsGiven $f(x)=2\cos \left(x-\frac{\pi}{3}\right)\cos x+1$, and $f\left(\alpha +\frac{\pi}{12}\right)=\frac{7}{6}$ with $\alpha \in \left(0,\frac{\pi}{2}\right)$, find $\sin \left(\frac{7\pi}{6}-2\alpha \right)$. Round 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.