Answer

Verified

438k+ 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

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Trending doubts

Difference Between Plant Cell and Animal Cell

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

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How do you graph the function fx 4x class 9 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

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