complex number Finding the Values of z for cos(z) = 2

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?

---

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.

Discover how to find the complex numbers z satisfying cos(z) = 2 with a detailed step-by-step solution. Explore Euler's formula, the quadratic equation formula, and the periodicity of the cosine function. Suitable for advanced mathematics enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation