Question

Find each of the following in the \(x+i y\) form and check your answers by computer. $$\cos (i \pi)$$

Short answer

Cos(iπ) = 11.5929 + 0i.

Step-by-step solution

Understanding the Problem

The problem requires finding the value of \(\text{cos}(i\text{π})\) in the form \(x + iy\).

Write the Euler's Formula for Cosine of a Complex Number

Euler's formula states that \(e^{iθ} = \text{cos}(θ) + i\text{sin}(θ)\). For a complex argument \(iπ\), we can derive the necessary formulas.

Apply Euler's Formula for Hyperbolic Functions

We use the fact that \( \text{cos}(iθ) = \text{cosh}(θ)\). In this case, \(θ = π\). Thus \( \text{cos}(i\text{π}) = \text{cosh}(π) \).

Compute the Hyperbolic Cosine

The hyperbolic cosine function is defined as \( \text{cosh}(x) = \frac{e^x + e^{-x}}{2} \). For \(x = π\), we get: \[ \text{cosh}(π) = \frac{e^{π} + e^{-π}}{2} \]

Final Evaluate to Get the Real Part

Evaluate \(\frac{e^{π} + e^{-π}}{2}\), which is a real number: e^{π} ≈ 23.1407 and e^{-π} ≈ 0.0432Thus \[ \text{cosh}(π) \approx \frac{23.1407 + 0.0432}{2} \approx 11.5929 \]

Write the Answer in the Form \( x + iy \)

Since \( \text{cos}(i\text{π}) = \text{cosh}(π) \) and it is a real number, the answer is \( \text{cos}(i\text{π}) = 11.5929 + 0i \).

Verify with a Computer

Use a computer or calculator to verify the result \( \text{cos}(i\text{π}) = 11.5929 \), confirming that our solution is correct.

Key concepts

Euler's Formula

Euler's formula is a fundamental bridge between exponential functions and trigonometry. It states that for any real number θ, the complex exponential is written as: \[ e^{iθ} = \text{cos}(θ) + i\text{sin}(θ) \]. This expression elegantly combines the real part \(\text{cos}(θ)\) and the imaginary part \(i\text{sin}(θ)\), providing insights that are widely used in both mathematics and engineering. When dealing with complex arguments, like in the exercise where we have \(iπ\), we must adapt this formula to handle the imaginary unit effectively.

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but for a hyperbola rather than a circle. They are especially useful in dealing with complex numbers:

• Hyperbolic cosine: \(\text{cosh}(x) = \frac{e^x + e^{-x}}{2}\)
• Hyperbolic sine: \(\text{ sinh}(x) = \frac{e^x - e^{-x}}{2}\)

In our problem, we use that \(\text{cos}(iθ) = \text{cosh}(θ)\). So, for \(θ = π\), \(\text{cos}(iπ)\) is equivalent to \(\text{cosh}(π)\). Understanding hyperbolic functions can clarify complex exponential calculations.

Cosine of Complex Numbers

Calculating the cosine of complex numbers combines Euler’s formula and hyperbolic functions. For a complex number \(z = ix\):

• We use \(\text{cos}(ix) = \text{cosh}(x)\)

This principle simplifies our exercise as: \[\text{cos}(iπ) = \text{cosh}(π)\]. Further computation requires the definition of hyperbolic cosine: \[\text{cosh}(π) = \frac{e^π + e^{-π}}{2} \]. Evaluating the values \(e^π ≈ 23.1407\) and \(e^{-π} ≈ 0.0432\) yields: \[\frac{23.1407 + 0.0432}{2} = 11.5929 \], demonstrating the intersection of these mathematical areas.

Real and Imaginary Parts

In the context of complex numbers, each number is expressed in terms of its real part and its imaginary part. A complex number \(z\) is written as \(z = x + iy\), where:

• \(x\) is the real part
• \(y\) is the imaginary part

For our exercise, we find \(\text{cos}(iπ)\) simplifies to the real number \(11.5929\). Thus, in the form \(x + iy\), the result is \(11.5929 + 0i\). Understanding how to separate into real and imaginary parts ensures clarity in complex number operations.