SOLUTION: The first three terms of an arithmetic progression are tan x, cos x, and sec x, respectively. If the k th term is cot x, find k.

Algebra ->  Algebra  -> Trigonometry-basics -> SOLUTION: The first three terms of an arithmetic progression are tan x, cos x, and sec x, respectively. If the k th term is cot x, find k.      Log On

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 Click here to see ALL problems on Trigonometry-basics Question 524900: The first three terms of an arithmetic progression are tan x, cos x, and sec x, respectively. If the k th term is cot x, find k.Answer by Aswathy(23)   (Show Source): You can put this solution on YOUR website!Note that AP means arithmetic progression. Here, a1=tanx, a2=cosx and a3=secx Common difference (d)of the given AP=a2-a1=cosx-tanx=cosx-(sinx/cosx) =(cos^2x-sinx)/(cosx)..............(1) Similarly,a3-a2=secx-cosx=(1/cosx)-cosx=(1-cos^2x)/cosx=(sin^2x)/cosx........(2) Since the given terms are in AP,then the difference between the terms will be equal. Therefore,a2-a1=a3-a2 then,cosx-tanx=secx-cosx cosx+cosx=secx+tanx 2cosx=(1/cosx)+(sinx/cosx) 2cosx=(1+sinx)/cosx 2cos^2x=1+sinx.................(3) Similarly,now from (1) and(2), (cos^2x-sinx)/(cosx)=(sin^2x)/(cosx) then,cos^2x-sinx=sin^2x {cancelling out cosx on both sides of the equation} cos^2x=sinx+sin^2x cos^2x=sinx(1+sinx) cos^2x=sinx(2cos^2x) [from (3)] cos^2x/2cos^2x=sinx 1/2=sinx {cancelling out cos^2x on both sides of the equation} Therefore,x=30 as sinx=1/2 Let ak(k is subscript of a) be the kth term which is cotx here. In the given AP,d=a2-a1=cos30-tan30=1/2sqrt(3) Now,kth term ,ak=a1+(k-1)d then,cos30=tan30+(k-1)2/2sqrt(3) By solving this equation we get,k=5 which is our required answer. So,k=5