Question

\(3\left(\cos 28^{\circ}+i \sin 28^{\circ}\right)\)

Short answer

The expression in exponential form is \( 3e^{i 28^{\text{°}}} \).

Step-by-step solution

Understand the Given Expression

The given expression is in the form of a complex number in polar form, specifically written as: \[ 3\big(\text{cos} 28^{\text{°}} + i \text{sin} 28^{\text{°}}\big) \]

Apply Euler’s Formula

Recall that Euler’s formula states: \[ e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta) \] Using this, the expression can be rewritten as: \[ 3 \times e^{i 28^{\text{°}}} \]

Switch to Rectangular Form (Optional)

If desired, convert the expression from polar form to rectangular form. Use \[ z = r(\text{cos}(\theta) + i \text{sin}(\theta)) \] Here, \( r = 3 \) and \( \theta = 28^{\text{°}} \), so the rectangular form is: \[ 3 \text{cos} 28^{\text{°}} + 3i \text{sin} 28^{\text{°}} \] Which numerically approximately equals: \[ 3 \times 0.8829 + 3i \times 0.4695 \ = 2.6487 + 1.4085i \]

Key concepts

Polar Form

The polar form of complex numbers is a unique and insightful way to represent complex numbers using a combination of their magnitude and angle.
This form reveals not only the magnitude or size of the complex number but also the direction or angle it makes with the positive real axis.
A complex number in polar form is written as: \( r(\text{cos}(\theta) + i \text{sin}(\theta)) \), where

• \( r \) is the magnitude (or modulus) of the number
• \( \theta \) is the angle made with the positive real axis (also known as the argument or phase)

Using our example, we have
\( r = 3 \) and \( \theta = 28^{\text{°}} \). This form is particularly useful in various branches of mathematics and engineering, such as signal processing, control theory, and quantum mechanics due to its simplicity in handling calculations involving multiplication and division of complex numbers.

Euler's Formula

Euler's formula is a stunning and profound connection between complex exponential functions and trigonometric functions.
It states: \( e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta) \), where:

• \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
• \( i \) is the imaginary unit (\( i^2 = -1 \))
• \( \theta \) is the angle in radians

Euler's formula makes it possible to represent complex numbers in an exponential form.
For a complex number in polar form \( 3(\text{cos} 28^{\text{°}} + i \text{sin} 28^{\text{°}}) \), we can apply the formula:
\( 3 \times e^{i 28^{\text{°}}} \).
This exponential form is compact and simplifies many operations involving complex numbers.
It is particularly advantageous in scenarios involving powers and roots of complex numbers.

Rectangular Form

The rectangular form is the standard form of representing complex numbers, in the form of \( a + bi \), where:

• \( a \) is the real part
• \( b \) is the imaginary part

This form directly provides the real and imaginary components of a complex number.
For instance, converting a polar form \( 3(\text{cos} 28^{\text{°}} + i \text{sin} 28^{\text{°}}) \) to a rectangular form results in:
\( 3 \text{cos} 28^{\text{°}} + 3i \text{sin} 28^{\text{°}} \).
Substituting the values, we get approximately:
\( 2.6487 + 1.4085i \).
The rectangular form is particularly useful for adding and subtracting complex numbers as it directly shows the component parts.