A flag pole of height ‘h’ metres is on the top of the hemispherical dome of radius ‘r’ metres. A man is standing 7 m away from the dome. Seeing the top of the pole at an angle 45° and moving 5 m away from the dome and seeing the bottom of the pole at an angle 30°. Find (i) the height of the pole (ii) radius of the (\( \sqrt { 3 } \) =1.732)

The angle of elevation of the top of a cell phone tower from the foot of a high apartment is 60° and the angle of depression of the foot of the tower from the top of the apartment is 30° . If the height of the apartment is 50 m, find the height of the cell phone tower. According to radiations control norms, the minimum height of a cell phone tower should be 120 m. State if the height of the above mentioned cell phone tower meets the radiation norms.

Three villagers A, B and C can see each other across a valley. The horizontal distance between A and B is 8 km and the horizontal distance between B and C is 12 km. The angle of depression of B from A is 20° and the angle of elevation of C from B is 30° . Calculate : the vertical height between A and B.(tan20° = 0.3640,(\(\sqrt { 3 } \) = 1.732)

If x sin³\(\theta \) + ycos³\(\theta \) = sin\(\theta \) cos\(\theta \) and x sin\(\theta \) = ycos\(\theta \), then prove that x² + y² = 1.

A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight and heads towards C on a bearing of 48° and distance 32 km away.
How far is B to the North of A? (sin 55° = 0.8192, cos 55° = 0.5736,sin 42° = 0.6691.cos 42° = 0.7431)

Two ships are sailing in the sea on either side of the lighthouse. The angles of depression of two ships as observed from the top of the lighthouse are 60° and 45° respectively. If the distance between the ships is 200\(\left( \frac { \sqrt { 3 } +1 }{ \sqrt { 3 } } \right) \) metres, find the height of the lighthouse.

prove that \({ \left( \frac { 1+sin\theta -cos\theta }{ 1+sin\theta +cos\theta } \right) }^{ 2 }=\frac { 1-cos\theta }{ 1+cos\theta } \)

if cot \(\theta \) + tan\(\theta \) = x and sec\(\theta \) - cos\(\theta \) = y, then prove that \(\begin{equation} \left(x^{2} y\right)^{\frac{2}{3}}-\left(x y^{2}\right)^{\frac{2}{3}}=1 \end{equation}\)

An Aeroplane sets of from G on bearing of 24° towards H, a point 250 km away, at H it changes  course and heads towards J deviates further by 55° and a distance of 180 km away.
How for is H to the east of G?

\(\left( \begin{matrix} sin24°=0.4067\quad sin11°=0.1908 \\ cos24°=0.9135\quad cos11°=0.9816 \end{matrix} \right) \)

As shown in the figure, two trees are standing on flat ground. The angle of elevation of the top of both the trees from a point X on the ground is 40° . If the horizontal distance between X and the smaller tree is 8 m and the distance of the top of the two trees is 20 m, calculate
(i) the distance between the point X and the top of the smaller tree.
(ii) the horizontal distance between the two trees.
(cos 40° = 0.7660)