Problem 29 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 29

Find standard notation, $a+b i$ $$2\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$$

Short Answer

Expert verified

The standard form is $0 + 2i$.

Step by step solution

Identify Euler's formula components

Euler's formula states that $ e^{i\theta} = \cos\theta + i\sin\theta $. Here, \theta = \frac{\pi}{2}.

Express in Euler's formula

Rewrite the given expression using Euler's formula: $ 2(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}) = 2 e^{i\frac{\pi}{2}} $.

Evaluate the trigonometric components

Evaluate the expression $e^{i\frac{\pi}{2}}$. This gives us: $e^{i\frac{\pi}{2}} = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} = i$ because $\cos\frac{\pi}{2} = 0$ and $\sin\frac{\pi}{2} = 1$.

Multiply by the scalar

Multiply the scalar 2 by the result from Step 3: $ 2 \cdot i = 2i $.

Write in standard form

Combine all parts: In standard form $a + bi$, $ a = 0 $ and $ b = 2 $, so the answer is $ 0 + 2i $.

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Key Concepts

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

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are in the form of $a + bi$. Here, $a$ and $b$ are real numbers, and $i$ is the imaginary unit. The imaginary unit is defined as $i = \sqrt{-1}$. So, for a complex number like $2 + 3i$:

$2$ is the real part.
$3i$ is the imaginary part.

Understanding complex numbers is crucial because they extend the idea of one-dimensional number lines to two dimensions. This is helpful for solving equations that involve the square roots of negative numbers, which don't have real solutions.

Trigonometric Functions

Trigonometric functions, such as sine ($\sin$) and cosine ($\cos$), relate angles to side lengths in right triangles. These functions are fundamental in geometry and complex numbers. For example:

$\cos\theta$ represents the adjacent side over the hypotenuse in a right triangle.
$\sin\theta$ represents the opposite side over the hypotenuse.

These functions are also essential for Euler's formula, which states: $e^{i\theta} = \cos\theta + i\sin\theta$. This relationship connects complex numbers with trigonometry and helps you convert between exponential and trigonometric forms of complex numbers easily.

Standard Notation

Standard notation for complex numbers is $a + bi$, where $a$ is the real part and $b$ is the coefficient of the imaginary part $i$. For instance, if you start with an expression like $2(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2})$, you can:

Rewrite it using Euler's formula.
Evaluate the trigonometric components.
Multiply any scalars involved.
Write the result in $a + bi$ form.

In our example:

$2(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2})$ becomes $2e^{i\frac{\pi}{2}}$,
$e^{i\frac{\pi}{2}}$ evaluates to $i$ (since $\cos\frac{\pi}{2} = 0$ and $\sin\frac{\pi}{2} = 1$),
Finally, multiplying by 2 gives us $2i$.

Expressing it in standard notation, we get $0 + 2i$.

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91影视

Find standard notation, \(a+b i\) $$2\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$$

Short Answer

Step by step solution

Identify Euler's formula components

Express in Euler's formula

Evaluate the trigonometric components

Multiply by the scalar

Write in standard form

Key Concepts

Complex Numbers

Trigonometric Functions

Standard Notation

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