Question

Find each of the following in rectangular form \(x+i y\) and check your results by computer. Remember to save time by doing as much as you can in your head. $$e^{-(i \pi / 4)+\ln 3}$$

Short answer

The rectangular form is \(\frac{3 \text{√2}}{2} - i \frac{3 \text{√2}}{2}\).

Step-by-step solution

- Use Euler's Formula

Express the given complex expression using Euler's Formula. Recall that Euler's formula states: \[e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta)\]Given expression: \[e^{-(i \frac{\text{π}}{4})+\text{ln} 3}\]Rewrite it as: \[e^{\text{ln}3} \times e^{-(i \frac{\text{π}}{4})}\]

- Simplify the Exponential Terms

Simplify each part individually.First, simplify the real exponent part: \[e^{\text{ln}3} = 3\]Next, simplify the complex exponent part using Euler's formula: \[e^{-(i \frac{\text{π}}{4})} = \text{cos}(-\frac{\text{π}}{4}) + i \text{sin}(-\frac{\text{π}}{4})\]

- Evaluate Trigonometric Functions

Evaluate the cosine and sine of the angle: \[\text{cos}(-\frac{\text{π}}{4}) = \text{cos}(\frac{\text{π}}{4}) = \frac{1}{\text{√2}} \text{ or } \frac{\text{√2}}{2}\]\[\text{sin}(-\frac{\text{π}}{4}) = -\text{sin}(\frac{\text{π}}{4}) = - \frac{1}{\text{√2}} \text{ or } - \frac{\text{√2}}{2}\]

- Multiply the Results to Get Rectangular Form

Combine the two results obtained:\[3 \times (\frac{\text{√2}}{2} - i \frac{\text{√2}}{2}) = \frac{3 \text{√2}}{2} - i \frac{3 \text{√2}}{2}\]Thus, the rectangular form is: \[x + iy = \frac{3 \text{√2}}{2} - i \frac{3 \text{√2}}{2}\]

Key concepts

Euler's formula

Euler's formula is a crucial concept in complex numbers. It links complex exponentials to trigonometric functions. Specifically, it states that for any real number \( \theta \), the complex exponential function can be written as: \[ e^{i \theta} = \cos(\theta) + i \sin(\theta) \] This formula allows us to transform expressions involving complex exponentials into sums of sine and cosine, which are often easier to work with. For instance, in our exercise, we used Euler's formula to rewrite \( e^{-(i \frac{\pi}{4})} \) in terms of cosine and sine. This made it straightforward to move to rectangular form.

Exponential functions

Exponential functions appear frequently in mathematics, representing processes that change exponentially. In the context of complex numbers, the exponential function \( e^{z} \) (where \( z \) is a complex number) can be expressed in terms of Euler's formula. This decomposition is powerful because it separates the problem into manageable parts.
In our exercise, we had the term \( e^{-(i \pi / 4) + \ln 3} \). We broke this down by isolating the real and imaginary parts:

• First, simplify \( e^{\ln 3} \) to 3, as the natural logarithm and the exponential function are inverses.
• Then, convert \( e^{-(i \pi/4)} \) using Euler's formula to \( \cos(-\pi/4) + i \sin(-\pi/4) \).

This approach made combining the parts easier, ultimately simplifying our expression.

Trigonometric functions

Trigonometric functions—cosine and sine—are essential in understanding waves, cycles, and rotations. Within complex numbers, they arise naturally from Euler's formula. In our exercise, we needed to evaluate these functions at a specific angle:

• \( \cos(-\pi/4) = \cos(\pi/4) = \frac{1}{\sqrt{2}} \) or \( \frac{\sqrt{2}}{2} \)
• \( \sin(-\pi/4) = -\sin(\pi/4) = -\frac{1}{\sqrt{2}} \) or \( -\frac{\sqrt{2}}{2} \)

Recognizing that cosine is an even function and sine is an odd function simplifies this evaluation. These values are substituted back into our equations, facilitating the transition to rectangular form.

Rectangular form

Complex numbers can be represented in different forms, and the rectangular form is particularly useful. It's written as \( x + iy \), where \( x \) and \( y \) are real numbers. In the exercise, after converting the exponential term using Euler's formula and simplifying the trigonometric components, we combined them into the rectangular form:

• Given: \( 3 \times ( \cos(-\pi/4) + i \sin(-\pi/4) ) \)
• Convert to: \( 3 \times ( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} ) \)
• Simplify to: \( \frac{3 \sqrt{2}}{2} - i \frac{3 \sqrt{2}}{2} \)

Thus, our final result in rectangular form is \( \frac{3 \sqrt{2}}{2} - i \frac{3 \sqrt{2}}{2} \). This process shows the power of these mathematical tools, making complex numbers easier to handle.