Problem 24 Find standard notation, \(a+b i\... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 8: Problem 24

Find standard notation, $a+b i$ $$6\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)$$

Short Answer

Expert verified

-3 + 3i\sqrt{3}

Step by step solution

Identify the trigonometric form

The given expression is in the trigonometric form of a complex number: \[6(\text{cos} 120^{\text{\circ}} + i \text{sin} 120^{\text{\circ}})\]

Convert cosine and sine to numerical values

Use trigonometric identities to find the values for $\text{cos} 120^{\text{\circ}}$ and $\text{sin} 120^{\text{\circ}}$:\[\text{cos} 120^{\text{\circ}} = -\frac{1}{2}\]\[\text{sin} 120^{\text{\circ}} = \frac{\sqrt{3}}{2}\]

Substitute trigonometric values into the expression

Replace the trigonometric functions with their numerical values in the expression:\[6 \left( -\frac{1}{2} + i \frac{\sqrt{3}}{2} \right)\]

Distribute the 6

Multiply both parts inside the parentheses by 6:\[6 \left( -\frac{1}{2} \right) + 6 \left( i \frac{\sqrt{3}}{2} \right)\]\[= -3 + 3 i \sqrt{3}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

standard notation of complex numbers

Complex numbers are a combination of a real number and an imaginary number. They are usually written in the form a + b i, where a and b are real numbers, and i is the imaginary unit with the property that i虏 = -1. This is known as the standard notation of complex numbers.
The real part of the complex number is a, and the imaginary part is bi. For example, in the complex number -3 + 3颈鈭�3, the real part is -3, and the imaginary part is 3颈鈭�3.
Understanding this notation is crucial because many operations with complex numbers, such as addition, subtraction, and multiplication, are based on treating the real and imaginary parts separately.

trigonometric identities

Trigonometric identities are equations involving the trigonometric functions: sine, cosine, and tangent, which are true for all values of the included variables. These identities are very useful in converting complex numbers from trigonometric form to standard notation.
In this exercise, we use the specific trigonometric values for 120 degrees:

cos(120掳) = -1/2
sin(120掳) = sqrt(3)/2

Using these values allows us to simplify the trigonometric form of the complex number and make the conversion to standard notation easier. Remembering key trigonometric values for common angles (like 30掳, 45掳, 60掳) is helpful for dealing with problems involving trigonometric identities.

complex number conversion

Converting complex numbers from trigonometric form to standard notation involves several steps. Here, we convert 6(cos(120掳) + i sin(120掳)) to a + bi form. The procedure is as follows:

1. Identify the trigonometric form.
2. Use trigonometric identities to find numerical values for cosine and sine.
3. Substitute these values into the expression:

cos(120掳) = -1/2
sin(120掳) = sqrt(3)/2

This gives us: 6(-1/2 + i sqrt(3)/2).

4. Distribute the 6 through the parentheses:

6 * (-1/2) + 6 * (i sqrt(3)/2) = -3 + 3颈鈭�3

Thus, the complex number in standard notation is -3 + 3颈鈭�3.

By breaking down the conversion process into these manageable steps, you can systematically approach and solve similar problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find standard notation, \(a+b i\) $$6\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)$$

Short Answer

Step by step solution

Identify the trigonometric form

Convert cosine and sine to numerical values

Substitute trigonometric values into the expression

Distribute the 6

Key Concepts

standard notation of complex numbers

trigonometric identities

complex number conversion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Discrete Mathematics

Geometry

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.