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Angle between Two Lines

At a point on...

Question

At a point on a level plane, a tower subtends an angle $α$ and a flag-staff of height $a$ metres standing on top of the tower subtends an angle $β$ . The height of the tower in metres is

\frac{a sin α sin (α + β)}{sin β}

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\frac{a cos α sin (α + β)}{sin β}

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\frac{a sin α cos (α - β)}{sin β}

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\frac{a sin α cos (α + β)}{sin β}

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Solution

The correct option is B $\frac{a sin α cos (α + β)}{sin β}$
$t a n α = \frac{h}{d}$
$t a n (α + β) = \frac{h + a}{d}$
$\frac{t a n α}{t a n (α + β)} = \frac{h}{h + a}$
$\frac{t a n (α + β)}{t a n α} = \frac{h + a}{h}$
$\frac{t a n (α + β) - t a n α}{t a n α} = \frac{a}{h}$
$h = \frac{a t a n α}{t a n (α + β) - t a n α}$
$h = \frac{a \frac{sin α}{cos α} . cos α cos (α + β)}{s i n (α + β) c o s α - c o s (α + β) s i n α}$
$h = \frac{a s i n α c o s (α + β)}{s i n β}$

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