$A$ ladder is leaning against a wall such that its upper end touches the wall at a height of $3 \, m$ and the ladder is inclined at an angle of $30^{\circ}$ with the ground. Find the length of the ladder in $m$.

The angle of elevation of the top of tower $N$ from the base of tower $M$ is $40^{\circ}$. The angle of elevation of the top of tower $M$ from the base of tower $N$ is $55^{\circ}$. Then,.......

Two trucks are parked in the same direction of a $300 \,m$ high tower. The angles of depression of these trucks from the top of the tower are found to be $45^{\circ}$ and $60^{\circ}$. Find the distance between these two trucks. (in $,m$)

From the top of a building $h \ m$ high,the angle of elevation of the top of a pole is found to be $\alpha$,while the angle of depression of the base of the pole is found to be $\beta$. Prove that the height of the pole is $h(1 + \tan \alpha \cdot \cot \beta) \ m$.

The angle of elevation of a cloud from a point $h$ metres above the surface of a lake is $\theta$ and the angle of depression of its reflection in the lake is $\phi$. Prove that the height of the cloud above the lake is $h\left(\frac{\tan \phi+\tan \theta}{\tan \phi-\tan \theta}\right).$

$A$ tower stands vertically on the ground. The angle of elevation of the top of the tower from a point $100 \, m$ away from the base of the tower is $60^\circ$. The height of the tower is $\ldots \ldots \ldots \ldots \, m$.