The angle of elevation of the sun,when the length of the shadow of a tree is $\sqrt{3}$ times the height of the tree,is (in $^{\circ}$)

The tops of two poles of height $60 \ m$ and $35 \ m$ are connected by a rope. If the rope makes an angle with the horizontal whose tangent is $\frac{5}{9}$,then what is the distance (in $m$) between the two poles?

$A$ $20 \, m$ high electric pole stands upright on the ground with the help of a steel wire attached to its top and fixed on the ground. If the steel wire makes an angle of $60^{\circ}$ with the horizontal ground,find the length of the steel wire (in $m$).

From the top of a cliff,$60 \ m$ high,the angles of depression of the top and bottom of a tower are observed to be $30^{\circ}$ and $60^{\circ}$ respectively. Find the height of the tower in $m$.

The angle of elevation of the top of a $TV$ tower from three points $A, B, C$ in a straight line through the foot of the tower are $\alpha, 2 \alpha, 3 \alpha$ respectively. If $AB = a$,the height of the tower is

If $x = a \cos \theta + b \sin \theta$ and $y = b \cos \theta - a \sin \theta,$ then $x^{2} + y^{2}$ is equal to