Answer

Verified

403.5k+ views

**Hint:**

The formula for writing tangent of an angle is

\[\tan \theta =\dfrac{perpendicular}{base}\] .

Another important formula that may be used in the question is the cosine rule which is

\[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-2ac\cos B\]

(Where a, b and c are the side lengths of the triangle and B is the angle opposite to the side length b)

**Complete step by step answer:**

As mentioned in the question, the figure would look like the below picture

(Where the height of the vertical pole is taken to be as ‘h’)

Let the distance between A and O be x and the distance between B and O be y.

Now, firstly using the tangent formula in triangle with \[\alpha \] angle as follows

\[\begin{align}

& \tan \alpha =\dfrac{h}{x} \\

& x=h\cot \alpha \ \ \ \ \ ...(1) \\

\end{align}\]

Secondly, using the tangent formula again in the triangle with \[\beta \] angle as follows

\[\begin{align}

& \tan \beta =\dfrac{h}{y} \\

& y=h\cot \beta \ \ \ \ \ ...(2) \\

\end{align}\]

Now, using the cosine formula in the \[\vartriangle AOB\]as follows

\[{{d}^{2}}={{x}^{2}}+{{y}^{2}}-2xy\cos \gamma \ \ \ \ \ ...(a)\]

Now, putting equations (1) and (2) in equation (a) as follows, we get

\[\begin{align}

& {{d}^{2}}={{h}^{2}}{{\cot }^{2}}\alpha +{{h}^{2}}{{\cot }^{2}}\beta -2(h\cot \alpha )(h\cot \beta )\cos \gamma \\

& {{d}^{2}}={{h}^{2}}{{\cot }^{2}}\alpha +{{h}^{2}}{{\cot }^{2}}\beta -2{{h}^{2}}\cdot \cot \alpha \cdot \cot \beta \cdot \cos \gamma \\

& {{d}^{2}}={{h}^{2}}\left( {{\cot }^{2}}\alpha +{{\cot }^{2}}\beta -2\cot \alpha \cdot \cot \beta \cdot \cos \gamma \right) \\

& {{h}^{2}}=\dfrac{{{d}^{2}}}{\left( {{\cot }^{2}}\alpha +{{\cot }^{2}}\beta -2\cot \alpha \cdot \cot \beta \cdot \cos \gamma \right)} \\

\end{align}\]

Now, on taking square root on both the sides, we get the height of the vertical pole that is standing on point O is

\[h=\dfrac{d}{{{\left( {{\cot }^{2}}\alpha +{{\cot }^{2}}\beta -2\cot \alpha \cdot \cot \beta \cdot \cos \gamma \right)}^{\dfrac{1}{2}}}}\]

**Note:**

The figure in this question is very tricky and is difficult to visualize it at first. Hence, the students can make an error while drawing the figure and then end up making a mistake and they would get to the correct solution. We can also solve the sides of the triangle using sine rule and then convert into cosine form.

Recently Updated Pages

If O is the origin and OP and OQ are the tangents from class 10 maths CBSE

Let PQ be the focal chord of the parabola y24ax The class 10 maths CBSE

Which of the following picture is not a 3D figure a class 10 maths CBSE

What are the three theories on how Earth was forme class 10 physics CBSE

How many faces edges and vertices are in an octagonal class 10 maths CBSE

How do you evaluate cot left dfrac4pi 3 right class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths