Solve the following equations for x a 2 tan x squareroot3

Solve the following equations for x a. 2 tan x - squareroot3 = tan x b. 4sin^2x + 1 = 4 c. cos x sin x = 3 cos x d. 2cos^2x - sinx - 1 = 0 e. sin 2x - sinx = 0

Solution

a)2tanx -3 =tanx

3tanx =3

tanx =3 /3

tanx =1/3

x =npi +(pi/3) , where n =0, 1,...

b)4sin²x +1 =4

4sin²x =3

sin²x =3/4

sinx =3 /2 , sinx =-3 /2

x=npi -(4pi/3) ,x=npi -(2pi/3)

c)cosxsinx=3cosx

cosxsinx-3cosx=0

cosx(sinx -3)=0

cosx=0

x =npi -(pi/2)

d)2cos²x -sinx -1=0

2(1-sin²x) -sinx -1=0

2-2sin²x -sinx -1=0

2sin²x +sinx -1=0

2sin²x +2sinx-sinx -1=0

2sinx(sinx+1) -1(sinx+1)=0

(sinx+1)(2sinx-1)=0

sinx=-1=>x=2npi-(pi/2)

sinx=1/2=>x=2npi +(pi/6),2npi+(5pi/6)

e)sin2x -sinx =0

2sinxcosx -sinx=0

sinx(2cosx -1)=0

sinx=0 =>x=npi

cosx =1/2=>x=2npi -(pi/3), 2npi +(pi/3)