Question

Write each complex number in rectangular form. $$ 2 e^{i \frac{\pi}{18}} $$

Short answer

1.97 + 0.348i

Step-by-step solution

Understand Euler's Formula

Recall Euler's formula which states: \[ e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta) \]In this problem, \( \theta \) is given as \( \frac{\theta}{18} \).

Apply Euler's Formula

Apply Euler's formula to \( 2 e^{i \frac{\theta}{18}} \): \[ 2 e^{i \frac{\theta}{18}} = 2 (\text{cos}(\frac{\theta}{18}) + i \text{sin}(\frac{\theta}{18})) \]

Calculate \( \text{cos}(\frac{\theta}{18}) \) and \( \text{sin}(\frac{\theta}{18}) \)

Compute the values of \( \text{cos}(\frac{\theta}{18}) \) and \( \text{sin}(\frac{\theta}{18}) \):\[ \text{cos}(\frac{\theta}{18}) = \text{cos}(\frac{\theta}{18}) \approx 0.985 \]\[ \text{sin}(\frac{\theta}{18}) = \text{sin}(\frac{\theta}{18}) \approx 0.174 \]

Multiply by 2

Multiply both the cosine and sine components by 2:\[ 2(\text{cos}(\frac{\theta}{18}) + i \text{sin}(\frac{\theta}{18})) = 2(0.985 + i 0.174) \]Simplify to get the rectangular form:\[ 2 \times 0.985 + 2 \times 0.174 i = 1.97 + 0.348i \]

Key concepts

Euler's Formula

Euler's formula is a fundamental bridge between complex numbers and trigonometry. It states that for any real number \(\theta\), the exponential form \(e^{i\theta}\) can be expressed as:
\[ e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta) \]
This relationship is incredibly useful in converting between different forms of complex numbers, making it easier to perform various mathematical operations. In our exercise, we use Euler's formula to convert the exponential form \(2 e^{i \frac{\pi}{18}}\) into its rectangular form by breaking it down into cosine and sine components.

Rectangular Form

The rectangular form of a complex number is expressed as \(a + bi\), where 'a' and 'b' are real numbers. In this form:

• 'a' represents the real part of the complex number.
• 'b' represents the imaginary part.

Our goal is to express the given complex number, which is currently in exponential form, into this standard rectangular form. By applying Euler's formula and calculating the cosine and sine components, we can achieve this conversion. In our specific problem, the calculations yield:
\[ 2 e^{i \frac{\pi}{18}} = 2(\text{cos}(\frac{\pi}{18}) + i \text{sin}(\frac{\pi}{18})) \]
By evaluating the trigonometric functions, we get: \[ 2(0.985 + i 0.174) = 1.97 + 0.348i \]

Cosine and Sine Functions

The cosine (\(\text{cos}\)) and sine (\(\text{sin}\)) functions form the core of trigonometry and are extensively used in dealing with angles and periodic phenomena.

• \(\text{cos}(\theta)\) provides the horizontal component of an angle \(\theta\).
• \(\text{sin}(\theta)\) gives the vertical component of the same angle.

In our example, the angle is \(\frac{\pi}{18}\). Calculating these functions:

• \(\text{cos}(\frac{\pi}{18}) \approx 0.985\)
• \(\text{sin}(\frac{\pi}{18}) \approx 0.174\)

We then multiply these values by 2 to determine the rectangular form:
\[ 2(\text{cos}(\frac{\pi}{18}) + i \text{sin}(\frac{\pi}{18})) \approx 2(0.985 + i 0.174) = 1.97 + 0.348i \] This final result is the complex number in rectangular form.

Multiplying Complex Numbers

Multiplying complex numbers involves distributing the multiplication over both the real and imaginary parts. For two complex numbers \((a + bi)\) and \((c + di)\), the multiplication is as follows:
\[ (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
Since \(i^2 = -1\), this simplifies to:
\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]
In our problem, after applying Euler's formula and calculating the cosine and sine values, multiplying by 2 adjusts both the real and imaginary components proportionally:

• Real part: \(2 \times 0.985 = 1.97\)
• Imaginary part: \(2 \times 0.174 = 0.348i\)

This multiplication solidifies our transformation of the complex number into its rectangular form.