An erect tree is $300\, m$ away from a hill. If the height of the hill is $300\, m$,find the angle of depression of the base of the tree from the top of the hill. (in $^{\circ}$)

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 shadow of a tower standing on a level plane is found to be $50 \, m$ longer when the Sun's elevation is $30^{\circ}$ than when it is $60^{\circ}$. Find the height of the tower (in $m$).

Watching from a point on the ground $20 \, m$ away from the base of an erect pole,the angle of elevation of the top of the pole is found to be $45^{\circ}$. Then,the height of the pole is $\ldots \ldots \ldots \, m$.

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

There is a flag-post on the top of a $10 \, m$ high tower. From a point on the ground,the angle of elevation of the top of the tower is $45^{\circ}$ and that of the top of the flag-post is $60^{\circ}$. Find the height of the flag-post. (in $, m$)