To find the complex number $z$ that satisfies $\cos(z) = 2$, we'll need to use the identity for the cosine of a complex number:

$\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$

Step 1: Set up the equation

Given: $\cos(z) = 2$

Substitute the expression for $\cos(z)$: $\frac{e^{iz} + e^{-iz}}{2} = 2$

Step 2: Multiply both sides by 2 to eliminate the fraction

$e^{iz} + e^{-iz} = 4$

Step 3: Set $e^{iz} = w$

Let $w = e^{iz}$. Then $e^{-iz} = \frac{1}{w}$, and the equation becomes: $w + \frac{1}{w} = 4$

Step 4: Multiply by $w$ to clear the fraction

$w^2 + 1 = 4w$

Step 5: Rearrange into a quadratic equation

$w^2 - 4w + 1 = 0$

Step 6: Solve the quadratic equation

Use the quadratic formula $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a = 1$, $b = -4$, and $c = 1$: $w = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}$ $w = \frac{4 \pm \sqrt{16 - 4}}{2}$ $w = \frac{4 \pm \sqrt{12}}{2}$ $w = \frac{4 \pm 2\sqrt{3}}{2}$ $w = 2 \pm \sqrt{3}$

Step 7: Find $z$ from $w$

Recall $w = e^{iz}$, so: $e^{iz} = 2 \pm \sqrt{3}$

Take the natural logarithm of both sides to solve for $z$: $iz = \ln\left(2 \pm \sqrt{3}\right)$ $z = -i \ln\left(2 \pm \sqrt{3}\right)$

Step 8: Include the general solution

The cosine function is periodic, so the general solution for $z$ will include $2n\pi$ (where $n$ is any integer): $z = -i \ln\left(2 \pm \sqrt{3}\right) + 2n\pi$

So, the general solutions for $z$ are: $z = -i \ln\left(2 + \sqrt{3}\right) + 2n\pi \quad \text{or} \quad z = -i \ln\left(2 - \sqrt{3}\right) + 2n\pi$

Would you like further clarification or have any questions about this solution?

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Here are 5 related questions to expand your understanding:

1. How do you solve similar trigonometric equations involving the sine function?
2. What is the general solution to $\sin(z) = 2$ for complex numbers?
3. How can you derive the complex logarithm of a number?
4. What are the properties of the cosine function when extended to complex numbers?
5. How do you find the inverse cosine of a complex number?

Tip: Remember that trigonometric functions for complex numbers involve both exponential and logarithmic functions, leading to solutions with multiple branches due to periodicity.