The approximate value of $\frac{1}{\sqrt{3}}$ up to $2$ decimal places is.........

The angle of elevation of the top of a tower from a certain point is $30^{\circ}$. If the observer moves $20 \, m$ towards the tower,the angle of elevation of the top increases by $15^{\circ}$. Find the height of the tower.

The angle of elevation of the top of a vertical tower from a point on the ground is $60^{\circ}$. From another point $10 \, m$ vertically above the first,its angle of elevation is $45^{\circ}$. Find the height of the tower.

In a right triangle,if the measure of one of the angles is $60^{\circ}$,then the measure of the side opposite to the angle with measure $60^{\circ}$ is $\ldots \ldots$ times the measure of the hypotenuse.

Write 'True' or 'False' and justify your answer.
If the height of a tower and the distance of the point of observation from its foot,both,are increased by $10 \%$,then the angle of elevation of its top remains unchanged.

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 $h\left(\frac{\tan \phi+\tan \theta}{\tan \phi-\tan \theta}\right).$