C

Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. $$\sqrt{3}\sin x=2-\cos x$$

The box is on the slope as in the picture. The length of the slope is $l=2\mathrm{m}$ and the height is $h=1.2\mathrm{m}$. The forces acting on the box are the force of gravity $\overrightarrow{F_G}$ and the friction $\overrightarrow{F_t}$. The force of gravity can be replaced by two components $\overrightarrow{F_1}$ and $\overrightarrow{F_n}$. (The force $\overrightarrow{F_1}$ is parallel to the slope and $\overrightarrow{F_n}$ is perpendicular to the slope.) The friction $F_t$ is given by the formula $F_t=f{F}_n$ where $f$ is the coefficient of the friction. Consider the standard acceleration of gravity $g=10\mathrm{m}{\mathrm{s}}^{-2}$. Find the minimal value for the coefficient of the friction $f$ to ensure that the box does not move with an acceleration.

The center of a spherical balloon is at a height of $500\mathrm{m}$ height. The visual angle of the balloon is $1^{\circ }{30}^{\prime }$. The elevation angle of the center of the balloon is ${42}^{\circ }{50}^{\prime }$. Find the diameter of the balloon in meters and round to nearest one decimal.

A tower is observed from two different places $A$ and $B$. The direct distance between $A$ and $B$ is $65\mathrm{m}$. If we denote the bottom of the tower by $C$, we get a triangle $ABC$ in which the measure of $\measuredangle CAB$ is ${71}^{\circ }$ and the measure of $\measuredangle ABC$ is ${34}^{\circ }$. From the point $A$ the angle of elevation of the top of the tower is ${40}^{\circ }{18}^{\prime }$. Find the height of the tower. Suppose that all $A$, $B$ and $C$ are in the same height above sea level and round your answers to nearest meters.