As observed from the top of a tower $30 \, m$ high,the angle of depression of a stone on the ground is found to be $45^{\circ}$. Find the distance between the stone and the tower (in $m$).

During a storm,a $21 \,m$ high tree breaks but the broken part remains attached to the trunk and makes an angle of $30^{\circ}$ with the ground. At what height from the ground is the tree broken? (in $m$)

In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $m \angle C = \theta$,then $\tan \theta = \dots$

Write 'True' or 'False' and justify your answer.
If the length of the shadow of a tower is increasing,then the angle of elevation of the sun is also increasing.

There is a $100\, m$ high tower at some distance from a cliff. The angle of elevation of the top of the cliff from the base of the tower is $45^{\circ}$ and the angle of depression of the base of the cliff from the top of the tower is $30^{\circ}$. Find the height of the cliff and also the distance between the tower and the cliff.

The angle of elevation of the top of a hill from a point on the ground is $30^\circ$. After walking $30 \, m$ towards the hill,the angle of elevation becomes $45^\circ$. What is the height of the hill? (in $m$)