Question

Cos(2x) = sin(x) Solve for the exact solutions in the interval [0,2pi]. If the equation has no solutions, respond with DNE.

Answer 1

The equation Cos(2x) = Sin(x) can be solved by using trigonometric identities and inverse functions. After transformations and factoring, we find the solutions to be x = π/6, 5π/6, and 3π/2 within the interval [0,2π].

Step-by-step explanation:

To solve the equation Cos(2x) = Sin(x) for exact solutions in the interval [0,2π], we can employ trigonometric identities and inverse functions. First, we can use a double-angle formula for cosine:

• Cos(2x) = 1 - 2Sin²(x)

By substituting this into our equation, we get:

1 - 2Sin²(x) = Sin(x)

Moving all terms to one side gives us a quadratic in terms of Sin(x):

2Sin²(x) + Sin(x) - 1 = 0

Now, we factor:

(2Sin(x) - 1)(Sin(x) + 1) = 0

Setting each factor equal to zero gives us two separate equations:

• 2Sin(x) - 1 = 0
• Sin(x) + 1 = 0

From there, we can find the solutions:

• x = Sin−1(1/2)
• x = Sin−1(−1)

Solving these will give us the solutions within the interval [0,2π]. If no solutions exist that satisfy the original equation within the given interval, we would respond with 'DNE'. However, in this case, we get the following solutions:

• x = π/6, 5π/6
• x = 3π/2

These are the exact solutions for the given interval.