Solve for x in sin xtan x + tan x – 2sin x + cos x = 0 for 0 ≤ x ≤2π rads.

Answer 1

Answer 2

sinxtanx +tanx -2sinx + cosx = 0

tanx = sinx/cosx

sinx(sinx/cosx) + sinx/cosx -2sinx + cosx =0

multiply by cosx to eliminate fractions

sin^2(x) + sinx -2sinxcosx +cos^2(x) = 0

replace 1st & last terms by 1, as sin^2(x)+cos^2(x) =1

sinx -2sinxcosx + 1 = 0

sinx(1-2cosx) = -1

1-2cosx =-1/sinx

-2cosx = -1/sinx -1

2cosx = 1/sinx +1

cosx = 1/2sinx + 1/2

sqr(1-sin^2(x)) = 1/2sinx + 1/2

square both sides

1-sin^2(x) = 1/4sin^2(x) + 1/2sinx + 1/4

multiply by 4sin^2(x)

4sin^2(x) -4sin^4(x) = 1 + 2sinx + sin^2(x)

4sin^4(x) -3sin^2(x) + 2sinx +1 = 0

let y = sinx

4y^4 -3y^2 +2y + 1 = 0

sinx = y = -1 or -0.3478

x = 270, 180-22.61 or -22.61 degrees

x= 270, 157.39, or 337.39 degrees

or

x = 3pi/2, 0.13pi, or 1.87pi radians

or

x= 4.71, 0.41, or 5.89 radians