cos2x(2cosx+1)=0

x=?

Either cos 2x is zero or cos x = -1/2.

Convert that to the appropriate angles x.

To solve the equation cos2x(2cosx+1) = 0 for x, we need to find the values of x that satisfy the equation. To do that, we can use the zero-product property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero.

In this equation, we have two factors: cos2x and (2cosx+1). We can set each factor individually equal to zero and solve for x.

1) cos2x = 0:

To find the values of x that satisfy this equation, we need to determine when the cosine of twice the angle, 2x, is equal to zero. The cosine function equals zero at every angle which is a multiple of π/2 radians, such as π/2, 3π/2, 5π/2, etc.

So we can write the first equation as:

2x = π/2 + kπ or 2x = 3π/2 + kπ

where k is an integer representing the number of complete cycles.

To solve for x, divide each equation by 2:

x = π/4 + (kπ)/2 or x = 3π/4 + (kπ)/2

2) (2cosx+1) = 0:

To find the values of x that satisfy this equation, we need to determine when 2cosx is equal to -1. Since the cosine function ranges from -1 to 1, this occurs when cosx = -1/2.

We can find the values of x by using the inverse cosine function (cos^(-1)):

x = cos^(-1)(-1/2) + 2kπ or x = -cos^(-1)(-1/2) + (2k+1)π

where k is an integer representing the number of complete cycles.

These are the solutions for x that satisfy the given equation cos2x(2cosx+1) = 0.