The angle of elevation of the top of the tower from a point $x \, m$ away from the tower is $30^{\circ}$. Then the height of the tower is $\ldots \ldots \ldots \, m$.

Write 'True' or 'False' and justify your answer.
$\tan \theta$ increases faster than $\sin \theta$ as $\theta$ increases for $0^\circ < \theta < 90^\circ$.

$A$ tree stands vertically on the bank of a river. The angle of elevation of the top of the tree from a point exactly opposite to the tree on the other bank of the river is $60^{\circ}$. The angle of elevation of the top of the tree from a point $40 \ m$ away from that point is $30^{\circ}$. Find the width of the river and the height of the tree.

$A$ person is watching from a point $A$ on the ground,which is $x \ m$ away from the base of a building. The height of the building is $y \ m$ and the angle of elevation of the top of the building is found to be $70^\circ$. Then:

The angle of depression of the top of a building from the top of a tower is $45^{\circ}$ and that of the base of the building from the top of the tower is $60^{\circ}$. If the height of the building is $7 \, m$,find the height of the tower.

'True' or 'False' and justify your answer.
If a man standing on a platform $3 \ m$ above the surface of a lake observes a cloud and its reflection in the lake,then the angle of elevation of the cloud is equal to the angle of depression of its reflection.