The angle of elevetion of the top of a tower from a point A in the north is α and the angle of the top of the tower from the point B in the east of the point A is β .prove that the height of the tower is ((ABsin αsin B)/( (√(sin^2 α−sin^2 β))))


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Question Number 177507 by peter frank last updated on 06/Oct/22

$The angle of elevetion of the$ $top of a tower from a point$ $A in the north is α and the$ $angle of the top of the tower$ $from the point B in the east of the point$ $A is β . prove that the height$ $of the tower is$ $\frac{ABsin α \sin B}{\sqrt{\sin^{2} α - \sin^{2} β}}$
	Answered by som(math1967) last updated on 06/Oct/22

	$l e t O P = h e i g h t = x$ $O A = x c o t α$ $O B = x c o t β$ $A B = \sqrt{{O B}^{2} - {O A}^{2}}$ $= \sqrt{x^{2} {c o t}^{2} β - x^{2} {c o t}^{2} α}$ $A B = x \sqrt{\frac{{c o s}^{2} β {s i n}^{2} α - {s i n}^{2} β {c o s}^{2} α}{{s i n}^{2} α {s i n}^{2} β}}$ $s i n α s i n β A B = x \sqrt{{s i n}^{2} α - {s i n}^{2} α {s i n}^{2} β - {s i n}^{2} β + {s i n}^{2} α {s i n}^{2} β}$ $∴ x = \frac{A B s i n α s i n β}{\sqrt{{s i n}^{2} α - {s i n}^{2} β}}$
	Commented by som(math1967) last updated on 06/Oct/22

	Commented by peter frank last updated on 06/Oct/22

	$thank you$
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