Problem 20 - Advanced Engineering Mathematics

Question

In Problems \(15-20\), find all values of \(z\) satisfying the given equation. $$ \cos z=i \sin z $$

Step-by-Step Solution

Answer

The values of \( z \) are of the form \( k\pi + \frac{\pi}{4} \), where \( k \) is an integer.

Step 1: Understand the Equation

The given equation is \( \cos z = i \sin z \). Recognize that this is a complex equation involving the cosine and sine of a complex number \( z \).

Step 2: Express in Exponential Form

Recall that for a complex number \( z = x + yi \), the exponential forms are \( e^{iz} = \cos z + i \sin z \) and \( e^{-iz} = \cos z - i \sin z \). The equation can be transformed using Euler's formulas by noting that \( \cos z = \frac{e^{iz} + e^{-iz}}{2} \) and \( \sin z = \frac{e^{iz} - e^{-iz}}{2i} \).

Step 3: Simplify the Equation

Use the fact that \( \cos z = \mathrm{Re}(e^{iz}) \) and \( i \sin z = i \cdot \mathrm{Im}(e^{iz}) \). This leads to \( \mathrm{Re}(e^{iz}) = i \cdot \mathrm{Im}(e^{iz}) \), which simplifies to \( e^{iz} + e^{-iz} = i \left(e^{iz} - e^{-iz}\right) \).

Step 4: Equate Real and Imaginary Parts

Consider \( e^w = \frac{1}{2}(\cos u - \sin u) + \frac{i}{2}(\cos u + \sin u) \) and equate real and imaginary parts: \( \cos u = \sin u \) and \( \sin u = - \cos u \), respectively. Simplify these equations to find points where these equalities hold.

Step 5: Solve for Trigonometric Function Equality

Solve the equation \( \cos u = i \sin u \) using trigonometric identities. Realize that this can be simplified to \( \sin 2u = 0 \) or \( 2u = n\pi \) where \( n \) is an integer, giving \( u = \frac{n\pi}{2} \).

Step 6: Determine Values of z

Based on \( u = \frac{n\pi}{2} \), set \( u = x + yi \) where \( z = x + yi \). We find that solutions occur at \( z = k\pi + \frac{\pi}{4} \) or any equivalent form where \( k \) is an integer.

Key Concepts

Euler's Formula

Euler's Formula is a fundamental bridge between trigonometry and complex numbers. It states that for any real number \( x \), the exponential function can be expressed as: \[ e^{ix} = \cos x + i\sin x \] This formula allows us to rewrite trigonometric expressions involving cosine and sine in terms of the exponential function. By analyzing equations in this form, we can more easily manipulate and solve them. Euler’s formula is particularly useful when dealing with complex equations, like the one given in our problem, because it simplifies the process of comparison and substitution.

• Transforms trigonometric identities into exponential form
• Simplifies calculations involving oscillations and waves
• Provides a more straightforward way to express rotations

Understanding Euler’s formula is essential for solving complex equations as it provides a cleaner way to handle and visualize multi-component trigonometric identities.

Complex Numbers

Complex numbers extend our number system to include a new dimension beyond the real numbers. They are represented in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit satisfying \( i^2 = -1 \). The beauty of complex numbers is that they allow us to perform mathematical operations that are not possible within the realm of real numbers alone, such as taking square roots of negative numbers.

• Used in solving equations that have no real solutions
• Crucial for fields like electrical engineering and fluid dynamics
• Enable representation of oscillations and rotations efficiently

In solving complex equations, we often convert expressions into complex number form to utilize their unique properties in simplifying and solving problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true. They are key tools for transforming and simplifying expressions involving angles. With regards to our equation, identities such as \( \cos u = \sin u \), and more complicated ones leveraged by Euler’s formula, help reduce complex expressions into more solvable forms. For instance, identities like \( \sin(2u) = 2\sin u \cos u \) allow us to simplify problems involving trigonometric equations. These identities are essential in solving the exercise, providing the necessary groundwork to equate parts and simplify the complex trigonometric equation.

• Help express one trigonometric function in terms of another
• Simplify difficult mathematical calculations
• Allow conversion between trigonometric and exponential forms

Mastery of trigonometric identities ensures efficient solving of complex equations by unlocking new pathways to solutions.

Real and Imaginary Parts

Distinguishing between the real and imaginary parts of a complex number is crucial for solving equations like the one given in the exercise. A complex number \( z = a + bi \) consists of:

• The real part \( a \)
• The imaginary part \( bi \)

In our exercise, converting the equation into exponential form using Euler’s formula allowed dissecting the complex sine and cosine into their real and imaginary components. This dissection facilitates solving, as we equate the separate real and imaginary parts to simplify and solve the equation. By comparing these components separately, the strategy is to simplify the equation, breaking it down to more manageable parts. This ultimately leads to determining the values of \( z \) that satisfy the complex equation, providing a clear path to finding solutions.