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Chapter 8: Problem 25

Find standard notation, \(a+b i\) $$10\left(\cos 270^{\circ}+i \sin 270^{\circ}\right)$$

Short Answer

Expert verified
The standard notation is \(-10i\).

Step by step solution

01

Identify the trigonometric values

Identify the values of \( \cos 270^{\circ} \) and \( \sin 270^{\circ} \). Knowing that \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \), substitute these values into the given expression.
02

Substitute the trigonometric values

Substitute \( \cos 270^{\circ} = 0 \) and \( \sin 270^{\circ} = -1 \) into the expression: \( 10(\cos 270^{\circ} + i \sin 270^{\circ}) = 10(0 + i(-1)) \). Simplify inside the parentheses to get \( 10(0 - i) \).
03

Simplify the expression

Multiply the 10 by each term inside the parentheses: \( 10(0 - i) = 10 \times 0 - 10 \times i = -10i \).
04

Write in standard form

Write the result in the standard form \(a + bi\). Since there is no real part, it would be \(0 - 10i\), which simplifies to \(-10i\).

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Key Concepts

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

Trigonometric Form of Complex Numbers
Complex numbers can be represented in different forms. One useful form is the trigonometric form. This form uses a radius (or modulus) and an angle (or argument). It is written as \( r(\text{cos}\theta + i\text{sin}\theta) \).
Here, \( r \) is the modulus, which is the distance from the origin to the complex number on the complex plane. The angle \( \theta \) is the argument, which is the angle formed with the positive x-axis.
For example, in the exercise, we have \( 10(\text{cos} 270^{\text{掳}} + i \text{sin} 270^{\text{掳}}) \).
Here, 10 is the modulus, and 270掳 is the argument.
Standard Form of Complex Numbers
Another way to write complex numbers is in the standard form, which is \( a + bi \). Here, \( a \) is the real part, and \( b \) is the imaginary part.
In this form, complex numbers are easy to add, subtract, and multiply.
To convert from trigonometric form to standard form:
  • Find the values of \( \text{cos} \theta \) and \( \text{sin} \theta \)
  • Multiply these values by the modulus \( r \)
  • Combine these to get \( a + bi \)
In our example, \( \text{cos} 270^{\text{掳}} = 0 \) and \( \text{sin} 270^{\text{掳}} = -1 \).
So, \( 10 (0 + i(-1)) = 10(0 - i) = -10i \). There is no real part here, so the standard form of the complex number is \( 0 - 10i \) or just \( -10i \).
Imaginary Unit
The imaginary unit is denoted by \( i \), and it is defined as \( i = \root{-1} \). This means that \(i^2\text{ = -1}\).
It is used to represent imaginary numbers, which are numbers that cannot be represented as real numbers.
In the standard form of complex numbers, the imaginary part always has a factor of \( i \).
For example, in the number \( -10i \), \( i \) indicates that it is an imaginary number, and -10 is the coefficient.
Imaginary numbers are useful in many fields, including engineering, physics, and applied mathematics. They allow us to solve equations that do not have real solutions.

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