Question

Find each of the following in rectangular form \(x+i y\). $$ e^{-(i \pi / 4)+\ln 3} $$

Short answer

\( \frac{3 \sqrt{2}}{2} - i \frac{3 \sqrt{2}}{2} \)

Step-by-step solution

Rewrite using Euler's Formula

Euler's formula states that any complex number in the form of an exponent can be rewritten as a product of a magnitude and a complex exponential: \[ e^{a + bi} = e^a \times e^{bi} \] In the given problem, rewrite \[ e^{-(i \pi / 4) + \ln 3} \] as \[ e^{\ln 3} \times e^{-(i \pi / 4)} \]

Simplify the Terms

Evaluate each term separately. The term \[ e^{\ln 3} \] simplifies to 3 because the exponential and logarithm are inverse functions: \[ e^{\ln 3} = 3 \] Next, we use Euler's formula for the term \[ e^{-(i \pi / 4)} \], which becomes: \[ e^{-(i \pi / 4)} = \cos(- \pi / 4) + i \sin(- \pi / 4) \]

Evaluate Cosine and Sine Functions

Find the values of the cosine and sine functions: \[ \cos(- \pi / 4) = \cos(\pi / 4) = \frac{\sqrt{2}}{2} \] \[ \sin(- \pi / 4) = - \sin(\pi / 4) = -\frac{\sqrt{2}}{2} \] Thus, \[ e^{-(i \pi / 4)} = \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \]

Multiply and Simplify

Combine the results of Steps 2 and 3: \[ e^{\ln 3} \times e^{-(i \pi / 4)} = 3 \times \left( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right) \] Distribute the 3: \[ 3 \times \frac{\sqrt{2}}{2} - 3i \times \frac{\sqrt{2}}{2} = \frac{3 \sqrt{2}}{2} - i \frac{3 \sqrt{2}}{2} \]

Key concepts

Euler's formula

To start, let's explore Euler's formula which is fundamental to complex numbers. Euler's formula is expressed as: \[ e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta) \]This equation creates a bridge between exponential functions and trigonometric functions. It allows expressing complex numbers in exponential form. For example, for any complex number in the form \[ e^{a+bi} = e^{a} \times e^{bi} \], we can split it into a magnitude part \[ e^a \] and an angle part \[ e^{bi} \].Understanding Euler's formula makes it easier to manipulate and convert forms of complex numbers. Beginners should note how it simplifies into combining cosines and sines, an elegant mix of two branches of mathematics.

Rectangular form

The rectangular form of a complex number is written as \[ x + iy \].It represents the complex number's real part (x) and imaginary part (y). Every complex number can be depicted as a point on the complex plane, with x as the coordinate on the real axis and y on the imaginary axis. For example, converting from exponential to rectangular form involves breaking down the components using trigonometric identities.In our exercise, we converted \[ e^{-(i \frac{\tissue{\text{latex}{}{}}{i} / 4) + \text{ln} 3} \] to a rectangular form by first simplifying \[ e^{\text{ln} 3} \] to 3, then using Euler's formula to express \[ e^{-(i \frac{\tissue{\text{latex}{}{}}{i} / 4)} \] as \[ \text{cos}(-\frac{\tissue{\text{latex}{}{}}{i}) / 4) + i \text{sin}(-\frac{\tissue{\text{latex}{}{}}{i}) / 4)} \]. Finally, combining these steps resulted in a rectangular form.

Exponential functions

Exponential functions are powerful tools in mathematics, e.g., \[ e^{x} \], where 'e' is Euler's number (approx. 2.718). They express growth or decay processes and operate seamlessly over complex numbers when paired with Euler's formula. For complex numbers, exponentiation allows moving between different representations.In our exercise, we restructured \[ e^{-(i \frac{\tissue{\text{latex}{}{}}{i} / 4) + \text{ln} 3} \] into separable exponentials, enabling us to handle the \[ e^{-\frac{\tissue{\text{latex}{}} / 4} \] part using trigonometric identities.

Trigonometric functions

Trigonometric functions like sine and cosine are key in converting between forms of complex numbers. In the context of Euler's formula, they define the multi-dimensional nature of complex numbers. For example, \[ \text{cos}(θ) + i \text{sin}(θ) \] translates a complex number from its magnitude and angle form into its rectangular coordinates.In the given problem, \[ \text{cos}(-\frac{\tissue{\text{latex}{}}{i_{\theta}} / 4}) \] and \[ \text{sin}(-\frac{\tissue{\text{latex}{}{}}{i_{\theta}} / 4}) \] simplified to familiar values like \[ \frac{sqrt2}{2} \] and \[ -\frac{sqrt2}{2} \], making the final steps straightforward to solve.