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Chapter 7: Problem 44

Write each complex number in the form \(a+b i\). $$0.5(\cos (5 \pi / 6)+i \sin (5 \pi / 6))$$

Short Answer

Expert verified
\( -\frac{\sqrt{3}}{4} + \frac{1}{4}i \)

Step by step solution

01

- Identify the components of the complex number

The given complex number is in polar form: \[0.5(\text{cos}(\frac{5\pi}{6}) + i \text{sin}(\frac{5\pi}{6})) \]. Here, 0.5 is the magnitude, \( \frac{5\pi}{6} \) is the argument, \(\cos\) represents the real part and \(\sin\) represents the imaginary part.
02

- Find the cosine and sine values

Calculate \(\cos(\frac{5\pi}{6})\) and \(\sin(\frac{5\pi}{6})\). We use unit circle values: \[ \cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2} \] \[ \sin(\frac{5\pi}{6}) = \frac{1}{2} \]
03

- Multiply by the magnitude

Multiply the real and imaginary parts by 0.5: \[ 0.5 \times \cos(\frac{5\pi}{6}) = 0.5 \times -\frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{4} \] \[ 0.5 \times \sin(\frac{5\pi}{6}) = 0.5 \times \frac{1}{2} = \frac{1}{4} \]
04

- Write in the form \(a + bi\)

Combine the real and imaginary parts to write the complex number in the form \(a + bi\): \[ -\frac{\sqrt{3}}{4} + \frac{1}{4}i \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

converting polar to rectangular form
Converting a complex number from polar to rectangular form is a multi-step process but very straightforward when broken down. In polar form, a complex number is represented as \(r(\text{cos}(\theta) + i\text{sin}(\theta))\), where \(r\) is the magnitude (or absolute value) and \(\theta\) is the argument (or angle). To convert from polar to rectangular form, you follow these steps:
  • Identify the magnitude (\(r\)) and angle (\(\theta\)).
  • Calculate the cosine and sine of the angle \(\theta\).
  • Multiply these values by the magnitude, \(r\).
  • Combine the real and imaginary parts to form \(a + bi\).
This method helps you transform and clearly understand the complex number in a new form.
complex numbers
A complex number is a number that consists of both a real part and an imaginary part. It can be written in the form \(a + bi\), where:
  • \(a\) is the real part
  • \(b\) is the imaginary part
  • \(i\) is the imaginary unit, equivalent to the square root of -1 (\(i = \frac{\text{-1}}{\text{}^{2}}\))
Complex numbers extend the idea of one-dimensional real numbers to two dimensions, using the real part and the imaginary part. They are crucial in many fields of science and engineering for depicting oscillations, waves, and other phenomena.
trigonometric functions in complex numbers
When dealing with complex numbers, trigonometric functions like \(\text{cos}\) and \(\text{sin}\) are essential. In the context of polar form, these functions describe the orientation of the number in the complex plane.
  • The \(\text{cos}\) function helps determine the real part.
  • The \(\text{sin}\) function helps determine the imaginary part.
For example, given \(\theta = \frac{5\frac{\text{\)\theta\(}}{\theta}}\) : \br>
  • \(\text{cos}(\frac{5\frac{\theta}{\theta}}) = \frac{{\theta}^{3}}\)
  • \(\text{sin}(\frac{5\frac{\theta}{\theta}}) = \frac{1{\theta}^{2}}\)
These values are then used to convert a complex number from polar to rectangular form by scaling them according to the magnitude \(r\).

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