Question

Find the exact solutions of the equation in the interval [0, 2π). sin(2x) cos(x) = 0

a) x = 0, π/2, π, 3π/2
b) x = π/2, π, 3π/2, 2π
c) x = 0, π, 2π
d) x = π/2, 3π/2

Answer 1

The exact solutions of the equation sin(2x) cos(x) = 0 in the interval [0, 2π) are x = 0, π/2, π, 3π/2.

Step-by-step explanation:

The equation sin(2x) cos(x) = 0 can be solved by setting each factor equal to zero and finding the values of x that satisfy each equation.

• sin(2x) = 0
• cos(x) = 0

For the equation sin(2x) = 0, we can set 2x = nπ, where n is an integer. Solving for x, we have x = nπ/2. The values of x that satisfy this equation in the interval [0, 2π) are x = 0, π/2, π, 3π/2.

For the equation cos(x) = 0, we can set x = (2n + 1)π/2, where n is an integer. The values of x that satisfy this equation in the interval [0, 2π) are x = π/2, 3π/2.

Combining the solutions for both equations, we get the exact solutions of the equation in the interval [0, 2π) as x = 0, π/2, π, 3π/2.