Sum of Trigonometric Ratios in Terms of Their Product
A tower has f...
Question
A tower has flag staff at its top which subtends equal angles α at a points distance 9 yds and 11 yds from the foot of the tower. If tanα=110, find the height of the tower and the flag staff.
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Solution
AB = y, tower, BC = x, flagstaff, AP = 9, AQ = 11. BC subtends equal angles α at P and Q s.t. tanα=110. Hence the circle through C and B must also pass through P and Q as angles in the same segment are equal. From the figure it is clear that segment BP subtends equal angles say β at C and Q. y+11tanβ ...(1) and x+y=11tan(α+β) ...(2) AP=9=(x+y)tanβ ...(3) ∴9=11tan(α+β)tanβ, from (2) and (3), or 9=11tanβ(tanα+tanβ1−tanαtanβ) Put tanα=110. ∴9[1−(−110)tanβ]=11tanβ(110+tanβ) 9(10−tanβ)=11tanβ(1+10tanβ). 110tan2β+20tanβ−90=0 or 11tan2β=2tanβ−9=0 (11tanβ−9)(tanβ+1)=0 ∴tanβ=911(βacute) Hence from (1), y=11tanβ=11.911=9 From (3), 9=(x+y)tanβ=(x+9).911. ∴x+9=11 or x=2.