Question

the angle of elevation of the top of a tower from two points 3 metre and 12 metre from the base of the tower and in the same straight line with it are complementary prove that the height of the tower is 6 metre​

Answer 1

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

• Let the height of tower AB be 'h' meter.

and

• Let C and D are two points on the same side of tower AB such that

• AC = 3 meter.

• AD = 12 meter.

• Let the angle of elevation from point C of the top of tower AB be 'x'

and

• Let the angle of elevation of point D of the top of tower AB be '90 - x.

Now,

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

$\rm :\longmapsto\:tanx \: = \: \dfrac{AB}{AC}$

$\rm :\longmapsto\:tanx \: = \: \dfrac{h}{ 3} - - - (1)$

Now,

$\rm :\longmapsto\:In \: \triangle \: BAD$

$\rm :\longmapsto\:tan(90 - x) = \dfrac{BA}{AD}$

$\rm :\longmapsto\:cotx \: = \: \dfrac{h}{12}$

$\rm :\longmapsto\:\dfrac{1}{tanx} = \dfrac{h}{12}$

$\rm :\longmapsto\:\dfrac{3}{h} = \dfrac{h}{12} \: \: \: \: \: \: \: \: \{ \because \: tanx \: = \: \dfrac{h}{3} \}$

$\rm :\longmapsto\: {h}^{2} = 36$

$\bf\implies \:h \: = \: 6 \: meter$

${\boxed{{\bf{Hence, Proved}}}}$

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