Question

Solve 2 cos² (x) + cos (x) = 1 on the interval [0, 2π). List exact radian answers if possible. If the exact answer is not possible, round answers to 3 decimal places.

Answer 1

To solve the equation 2 cos²(x) + cos(x) = 1 on the interval [0, 2π), rewrite it as a quadratic equation by letting cos(x) = y. Factor the quadratic equation (2y - 1)(y + 1) = 0. Solve for x by taking the inverse cosine of both values of y.

Step-by-step explanation:

To solve the equation 2 cos²(x) + cos(x) = 1 on the interval [0, 2π), we can rewrite it as a quadratic equation by letting cos(x) = y.

Thus, the equation becomes 2y² + y - 1 = 0.

We can now factor this quadratic equation as (2y - 1)(y + 1) = 0.

Therefore, y = 1/2 or y = -1.

Now, we can solve for x by taking the inverse cosine of both values of y.

For y = 1/2,

we have cos(x) = 1/2.

This occurs when x = π/3 or

x = 5π/3.

For y = -1,

we have cos(x) = -1.

This occurs when x = π.