Solve for x in (0 ; 360) : 2 tan x - 1 = -tan (-x).
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We have to solve 2 tan x - 1 = -tan (-x)
Now we know that tan -x = - tan x
2 tan x - 1 = -tan (-x)
=> 2 tan x - 1 = tan (x)
=> tan x - 1 = 0
=> tan x = 1
=> x = arc tan 1
=> x = 45 degrees and 225 degrees.
The required values of x = 45 degrees and x = 225 degrees
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First, we'll re-write the right side term, based on the fact that the tangent function is odd, so tg(-x)=-tgx.
The equation will become:
2tanx-1=-(-tan x)
We'll remove the brackets:
2tan x-1 = tan x
We'll subtract tan x both sides:
2tan x-1 - tan x = 0
We'll combine lie terms:
tan x - 1 = 0
We'll add 1:
tan x =1
x=arctan1 + k*pi
But arctan 1= pi/4
x = pi/4
The tangent is also positive in the 3rd quadrant, so x = pi + pi/4
x = 5pi/4
The possible values of the angle are {pi/4 ; 5pi/4}.
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