Find the exact solutions of the equation in the interval [0, 2π).
(cos 2x) - (cos x)=0
cos (2x) - cosx = 0
2cos^2 x - 1 - cosx = 0
(2cosx + 1)(cosx - 1) = 0
cosx = -1/2 or cosx = 1
if cosx = -1/2, then
x = 120° or 240°
x = 2π/3 or x = 4π/3
if cosx = 1
x = 0 or x = 2π
x = 0, 2π/3 , 4π/3 , 2π
How did you factor the 2 cos^2x - 1 - cosx?
2cos^2 x - cosx - 1
compare to
2a^2 - a - 1
= (2a +1)(a - 1) , where a = cosx
Ah, okay. Thank you!
To find the exact solutions of the equation (cos 2x) - (cos x) = 0 in the interval [0, 2π), we can use trigonometric identities and algebraic manipulation.
1. Start by applying the double angle formula for cosine:
cos 2x = 2(cos^2 x) - 1
2. Substitute the double angle formula into the equation:
2(cos^2 x) - 1 - (cos x) = 0
3. Rearrange the equation:
2(cos^2 x) - (cos x) - 1 = 0
4. Now we can solve this quadratic equation for cos x. Let's substitute cos x with a variable to simplify the notation. Let's say u = cos x. Then the equation becomes:
2(u^2) - u - 1 = 0
5. Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we will use the quadratic formula:
u = [-b ± √(b^2 - 4ac)] / (2a)
For our equation, a = 2, b = -1, and c = -1. Plugging these values into the quadratic formula, we get:
u = [1 ± √((-1)^2 - 4(2)(-1))] / (2(2))
= [1 ± √(1 + 8)] / 4
= [1 ± √9] / 4
= [1 ± 3] / 4
6. Simplifying further, we have two possible values for u:
u₁ = (1 + 3) / 4 = 4 / 4 = 1
u₂ = (1 - 3) / 4 = -2 / 4 = -1/2
7. Now, we need to find the values of x that correspond to these values of u. Recall that we substituted u = cos x. Since the cosine function has a range of [-1, 1], we only need to consider values of x that give us u equal to or between -1 and 1.
For u = 1, we have cos x = 1, which means x = 0. This solution falls within the interval [0, 2π).
For u = -1/2, we have cos x = -1/2. By using the inverse cosine function (or cosine table), we can find the angles corresponding to this cosine value. The solutions are x = 2π/3 and x = 4π/3. Both of these solutions also fall within the interval [0, 2π).
Therefore, the exact solutions of the equation (cos 2x) - (cos x) = 0 in the interval [0, 2π) are x = 0, x = 2π/3, and x = 4π/3.