We are given the equation $\cos^2(x) - 4\cos(x) - 1 = 0$ and we need to find all solutions in the interval $[0, 2\pi)$.

Step 1: Let $u = \cos(x)$

We can rewrite the equation as a quadratic equation in terms of $u$: $u^2 - 4u - 1 = 0$

Step 2: Solve the quadratic equation

To solve the quadratic equation, we use the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -4$, and $c = -1$. Plugging in these values, we get: $u = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)}$ $u = \frac{4 \pm \sqrt{16 + 4}}{2}$ $u = \frac{4 \pm \sqrt{20}}{2}$ $u = \frac{4 \pm 2\sqrt{5}}{2}$ $u = 2 \pm \sqrt{5}$

Step 3: Analyze the solutions

Now, we have two potential solutions for $u$: $u_1 = 2 + \sqrt{5}, \quad u_2 = 2 - \sqrt{5}$ However, since $u = \cos(x)$ and the cosine of an angle is always between $-1$ and $1$, we discard $u_1 = 2 + \sqrt{5}$ because it is greater than 1. Thus, we only consider: $u = 2 - \sqrt{5}$

Step 4: Find the corresponding angle $x$

To find $x$, we solve: $\cos(x) = 2 - \sqrt{5}$ Since the cosine function is even, $x$ will have two solutions within the interval $[0, 2\pi)$: $x_1 = \cos^{-1}(2 - \sqrt{5})$ $x_2 = 2\pi - \cos^{-1}(2 - \sqrt{5})$

Step 5: Calculate the solutions

Let's calculate the approximate values for $x_1$ and $x_2$ in radians:

$x_1 \approx \cos^{-1}(2 - \sqrt{5}) \approx \cos^{-1}(-0.2361) \approx 1.8091 \, \text{radians}$ $x_2 \approx 2\pi - 1.8091 \approx 4.4741 \, \text{radians}$

Final Answer

The solutions in the interval $[0, 2\pi)$ are: $\boxed{1.8091 \, \text{radians}, \, 4.4741 \, \text{radians}}$

Would you like a detailed explanation of any step?

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Here are some related questions you might explore:

1. What are the general solutions for the equation $\cos^2(x) - 4\cos(x) - 1 = 0$?
2. How would you solve $\sin^2(x) - 4\sin(x) - 1 = 0$ in the same interval?
3. How can you verify the solutions to the equation using a graph?
4. What is the derivative of the function $f(x) = \cos^2(x) - 4\cos(x) - 1$?
5. How would you find the critical points of the function $f(x) = \cos^2(x) - 4\cos(x) - 1$?
6. How can you solve the equation using numerical methods instead of the quadratic formula?
7. What are the key differences between solving trigonometric equations and algebraic equations?
8. How can you use trigonometric identities to simplify similar equations?

Tip: Always remember to check if your solutions are within the given interval, especially when dealing with trigonometric functions.