Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 5

Chapter 5, Problem 5

In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)

Verified step by step guidance

Identify the expression to simplify: \(6 - (-5 + 4i) - (-13 - i)\).

Apply the distributive property to remove the parentheses by changing the signs inside each parenthesis preceded by a minus: \(6 + 5 - 4i + 13 + i\).

Group the real parts together and the imaginary parts together: \((6 + 5 + 13) + (-4i + i)\).

Add the real parts: \(6 + 5 + 13\), and add the imaginary parts: \(-4i + i\) separately.

Write the final expression in standard form \(a + bi\), where \(a\) is the sum of the real parts and \(b\) is the sum of the imaginary parts.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. Writing a complex number in standard form means presenting it clearly as a sum or difference of its real and imaginary components.

Addition and Subtraction of Complex Numbers

To add or subtract complex numbers, combine their real parts separately and their imaginary parts separately. This process is similar to combining like terms in algebra, ensuring the result remains in standard form.

Handling Negative Signs and Parentheses

When subtracting complex numbers, carefully distribute the negative sign across all terms inside the parentheses. This step is crucial to avoid sign errors and correctly simplify the expression.

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In Exercises 29–36, simplify and write the result in standard form.

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In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/12 + i sin π/12)]⁶

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In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√(−16) + 3√(−81)

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