Question

use Euler’s formula to write the given expression in the form a + ib. $$ e^{2-(x / 2) i} $$

Short answer

Question: Rewrite the expression \(e^{2-(x/2)i}\) in the form of \(a + ib\). Answer: \(\cos(2-\frac{x}{2}) + i\sin(2-\frac{x}{2})\).

Step-by-step solution

Apply Euler's formula

First, we apply Euler's formula to the given expression, where \(x = 2 - \frac{x}{2}\). So we have $$ e^{(2-\frac{x}{2})i} = \cos(2-\frac{x}{2}) + i\sin(2-\frac{x}{2}). $$

Simplify the expression

We can now rewrite the expression in the form \(a + ib\) by identifying the real and imaginary parts. The real part, \(a\), is given by \(\cos(2-\frac{x}{2})\), and the imaginary part, \(b\), is given by \(\sin(2-\frac{x}{2})\). So the final expression is: $$ a + ib = \cos(2-\frac{x}{2}) + i\sin(2-\frac{x}{2}). $$ Therefore, the given expression \(e^{2-(x/2)i}\) can be written in the form \(a + ib\) as: \(\cos(2-\frac{x}{2}) + i\sin(2-\frac{x}{2})\).

Key concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics, especially when dealing with problems that extend beyond the real number line. A complex number is typically expressed in the form \(a + ib\), where:

• \(a\) is the real part
• \(b\) is the imaginary part
• \(i\) is the imaginary unit (where \(i^2 = -1\))

Complex numbers allow for the representation of numbers that have both a real and an imaginary component, opening possibilities for solving equations where square roots of negative numbers occur.
In our exercise, we learned how Euler's formula helps transform an exponential expression with imaginary components into a complex number with easily identifiable real and imaginary parts.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are crucial in understanding Euler's formula. They help relate complex exponentials to their equivalent expressions in terms of complex numbers.
In the formula \(e^{ix} = \cos(x) + i \sin(x)\), these functions describe the position of a point on the unit circle:

• \(\cos(x)\) determines the horizontal coordinate, representing the real part of the complex number.
• \(\sin(x)\) determines the vertical coordinate, representing the imaginary part.

This relationship also underscores the periodic nature of trigonometric functions, making them indispensable in applications involving rotations and oscillatory phenomena.
In our solution, expressing \(e^{(2-\frac{x}{2})i}\) in terms of sine and cosine allowed us to easily rewrite it in the standard complex form of \(a + ib\).

Exponential Form

The exponential form of complex numbers is another powerful tool for solving and simplifying complex number expressions. Through Euler's formula, \(e^{ix} = \cos(x) + i \sin(x)\), one can represent complex numbers as exponentials, which is often more convenient for multiplication and division.

This form is particularly useful because it allows us to transform intricate trigonometric expressions into a more manageable form. When you encounter a complex number in exponential form, you can quickly see its effect in calculations, such as rotations, by manipulating just the exponent.

• The base \(e\) (the natural exponential) lends beneficial properties.
• The imaginary unit \(i\) introduces a phase rotation effect due to its trigonometric equivalence.

In our exercise, recognizing the exponential part \(e^{(2-\frac{x}{2})i}\) was pivotal in leveraging Euler's formula to express the original complex number in an easily understood \(a + ib\) form.