Question

A boy standing on a horizontal plane finds a bird flying at a distance of 100 m from him at an angle
of elevation of 30°. A girl standing on the roof of 20 m high building finds the angle of elevation of the
same bird to be 45°. Both the boy and the girl are on opposite side of the bird. Find the distance of the
bird from the girl.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let

• A be the position of the bird.

• B be the position of Boy on the ground.

• D be the position of the girl

• ED be the height of building at which girl is standing.

So,

We have,

• AB = 100 m

• Height of building, ED = 20 m

Now,

$\rm :\longmapsto\:In \: \triangle \: ABC$

$\rm :\longmapsto\:sin30 \degree \: = \dfrac{AC}{AB}$

$\rm :\longmapsto\:\dfrac{1}{2} = \dfrac{AC}{100}$

$\bf\implies \:AC \: = \: 50 \: m$

Now,

• DE = FC = 20 m

Also,

• AC = AF + FC

• AF = AC - FC

• AF = 50 - 20

• AF = 30 m.

$\rm :\longmapsto\:In \: \triangle \: AFD$

$\rm :\longmapsto\:sin45 \degree \: = \dfrac{AF}{AD}$

$\rm :\longmapsto\: \dfrac{1}{ \sqrt{2} } \: = \dfrac{30}{AD}$

$\rm :\longmapsto\:AD = 30 \sqrt{2}$

$\rm :\longmapsto\:AD = 30 \times 1.41$

$\rm :\longmapsto\:AD = 42.3 \: m$

Additional Information :-

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$