Final answer:
The exact radian solutions to the equation sinx + 2sinxcosx = 0 are x = 0, x = π/3, x = π, and x = 5π/3.
Step-by-step explanation:
To find all exact radian solutions to the equation sinx + 2sinxcosx = 0, we can factor out the common term sinx, giving us sinx(1 + 2cosx) = 0. This equation is satisfied when either sinx = 0 or 1 + 2cosx = 0.
If sinx = 0, the solutions are x = 0 and x = π.
If 1 + 2cosx = 0, we can solve for cosx to find cosx = -1/2. The solutions to this equation are x = π/3 and x = 5π/3.
Therefore, the exact radian solutions to the equation are x = 0, x = π/3, x = π, and x = 5π/3.