Final answer:
To solve the equation cos(2x) = sin(x) in the interval [0,2pi), rewrite it using trigonometric identities to get 2sin(x) = 1, and divide both sides by 2 to get sin(x) = 1/2. The exact solutions for sin(x) = 1/2 in the interval [0,2pi) are x = pi/6 and x = 5pi/6.
Step-by-step explanation:
To solve the equation cos(2x) = sin(x) in the interval [0,2pi), we need to rewrite the equation using trigonometric identities. We know that sin(2x) = 2sin(x)cos(x), so we can rewrite the equation as 2sin(x)cos(x) = sin(x). Now we can solve for sin(x) by dividing both sides of the equation by cos(x), giving us 2sin(x) = 1. Dividing both sides by 2, we get sin(x) = 1/2.
In the given interval [0,2pi), the exact solutions for sin(x) = 1/2 are x = pi/6 and x = 5pi/6.