double angle applicationcalculate cot 2x if cos x=(-sin x)/2

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to find cot 2x give that cos x=(-sin x)/2

cot x = cos x / sin x

tan x = 1/cot x

tan 2x = 2*tan x/ (1 - (tan x)^2)

we have cos x=(-sin x)/2

=> sin x /cos x = -2

=> tan x = -2

tan 2x = 2*(-2)/(1 - 4)

=> -4/-3

=> 4/3

cot 2x = 3/4

The required value of cot 2x is 3/4

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll re-write the expresison:

2cosx = -sinx

2cos x + sin x = 0

If we'll divide the constraint from enunciation by cos x, the expression will become:

2 + sin x/cos x = 0

We'll subtract 2:

sin x/cos x = -2

But sin x/cos x = tan x

tan x = -2

We'll write tan 2x:

tan 2x=tan (x+x)

tan 2x = (tan x+ tan x)/[1-(tan x)^2]

tan 2x =  2tan x/[1-(tan x)^2]

We'll substitute tan x = -2

tan 2x =  2*(-2)/[1-(-2)^2]

tan 2x = -4/(1-4)

tan 2x = -4/-3

tan 2x = 4/3

But cot 2x = 1/tan 2x = 3/4.

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