tan(x) − 2 sin(x) tan(x) = 0 Find all exact solutions on [0, 2π)

Answer 1

The previous answer correctly listed 2 solutions, namely: x=π/6 and x=5π/6. More trivial solutions x=0 and x=π should be added to the list because of the "inclusive left bracket" in problem description: [0, 2π)

Answer 2

Factor the equation:

tan(x)[1 - 2sin(x)] = 0

Using the zero product rule, set each of the factors equal to zero to solve

a) tan(x) = 0 which has the solution x = 0 or x = π on the interval given

b) 1 - 2sin(x) = 0 or sin(x) = 1/2 which has the solution x = π/6 and x = 5π/6 on the interval given

So the solutions are x = 0, π/6, 5π/6, and π

Answer 3

Tan(x) - 2tan(x)sin(x)=0 Factor out tan x

tanx(1-2sinx)=0

tan x=0 at 0,pi and2pi

1-2sinx=0

-2sinx=-1

sinx=1/2 at pi/6 and 5pi/6