The given expression is \( Z = \frac{1 + it}{1 - it} \), where \( t = \tan \frac{2\pi}{5} \). Let's compute \( Z \) step by step.

### Step 1: Simplify the expression for \( Z \)

We need to simplify the given expression. To do this, we multiply both the numerator and the denominator by the complex conjugate of the denominator, \( (1 + it) \):

\[
Z = \frac{(1 + it)(1 + it)}{(1 - it)(1 + it)} = \frac{(1 + it)^2}{1^2 - (it)^2}
\]

### Step 2: Simplify the denominator

The denominator is a difference of squares:

\[
1^2 - (it)^2 = 1 - (-t^2) = 1 + t^2
\]

### Step 3: Expand the numerator

Now, expand \( (1 + it)^2 \):

\[
(1 + it)^2 = 1^2 + 2 \cdot 1 \cdot it + (it)^2 = 1 + 2it - t^2
\]

Thus, we have:

\[
Z = \frac{1 + 2it - t^2}{1 + t^2}
\]

### Step 4: Plug in the value of \( t = \tan \frac{2\pi}{5} \)

The value of \( \tan \frac{2\pi}{5} \) is approximately:

\[
t = \tan \frac{2\pi}{5} \approx 0.7265
\]

Now substitute \( t \) into the expression for \( Z \):

\[
Z = \frac{1 + 2i(0.7265) - (0.7265)^2}{1 + (0.7265)^2}
\]

### Step 5: Compute the terms

First, calculate \( t^2 \):

\[
t^2 = (0.7265)^2 \approx 0.5278
\]

Now substitute this into the expression for \( Z \):

\[
Z = \frac{1 + 2i(0.7265) - 0.5278}{1 + 0.5278} = \frac{1 - 0.5278 + 1.453i}{1.5278}
\]

This simplifies to:

\[
Z = \frac{0.4722 + 1.453i}{1.5278}
\]

Now, divide each term by \( 1.5278 \):

\[
Z \approx 0.309 + 0.951i
\]

### Final Answer:

\[
Z \approx 0.309 + 0.951i
\]

This is the simplified form of \( Z \) when \( t = \tan \frac{2\pi}{5} \).

Would you like more details or have any questions? Here are 5 related questions you might find helpful:

1. How does multiplying by the complex conjugate help simplify a complex fraction?
2. What are the properties of the tangent function at specific angles like \( \frac{2\pi}{5} \)?
3. Can we represent complex numbers in polar form for a different perspective on \( Z \)?
4. How can we find the argument and modulus of a complex number like \( Z \)?
5. How do operations with complex numbers work geometrically?

**Tip:** Multiplying by the complex conjugate is useful when simplifying complex fractions because it removes the imaginary part from the denominator.

Z = (1 + it) / (1 - it) if t = tan(2π/5)

Explore the step-by-step process of simplifying the complex expression Z = (1 + it) / (1 - it) where t = tan(2π/5). This detailed solution covers key concepts in complex numbers and trigonometry, including the use of the complex conjugate and special trigonometric values. Suitable for advanced university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation