Giải phương trình lượng giác

To solve the equation 2sin^3x + cos2x - cosx = 0, we can use some trigonometric identities.

First, let's rewrite cos2x in terms of sinx using the identity cos2x = 1 - 2sin^2x:

2sin^3x + (1 - 2sin^2x) - cosx = 0

Next, let's combine like terms:

2sin^3x - 2sin^2x - cosx + 1 = 0

Now, let's factor out sinx:

sinx(2sin^2x - 2sinx - 1) + 1 = 0

Now, we have a quadratic equation in terms of sinx. Let's solve it:

2sin^2x - 2sinx - 1 = 0

Using the quadratic formula, we have:

sinx = (2 ± √(4 - 4(2)(-1))) / (2(2))
sinx = (2 ± √(4 + 8)) / 4
sinx = (2 ± √12) / 4
sinx = (2 ± 2√3) / 4
sinx = (1 ± √3) / 2

So, the possible values for sinx are (1 + √3) / 2 and (1 - √3) / 2.

Now, let's find the corresponding values for cosx using the identity cosx = √(1 - sin^2x):

cosx = √(1 - ((1 + √3) / 2)^2) = √(1 - (1 + 2√3 + 3) / 4) = √(1 - (4 + 2√3 + 3) / 4) = √(1 - (7 + 2√3) / 4) = √((4 - 7 - 2√3) / 4) = √((-3 - 2√3) / 4) = √(-3 - 2√3) / 2

cosx = √(1 - ((1 - √3) / 2)^2) = √(1 - (1 - 2√3 + 3) / 4) = √(1 - (4 - 2√3 + 3) / 4) = √(1 - (7 - 2√3) / 4) = √((4 - 7 + 2√3) / 4) = √((-3 + 2√3) / 4) = √(-3 + 2√3) / 2

Therefore, the solutions to the equation 2sin^3x + cos2x - cosx = 0 are:

sinx = (1 + √3) / 2, cosx = √(-3 - 2√3) / 2

and

sinx = (1 - √3) / 2, cosx = √(-3 + 2√3) / 2