Textbook Question
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
x² + y² = 16
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.1.47
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.1.47Chapter 5, Problem 5.1.47
In Exercises 37–52, perform the indicated operations and write the result in standard form.
( −6 − √(−12)) / 48
Verified step by step guidance1
Identify the expression to simplify: \(\frac{-5 - \sqrt{-12}}{48}\). Notice that \(\sqrt{-12}\) involves an imaginary number because of the negative under the square root.
Rewrite the square root of the negative number using the imaginary unit \(i\), where \(i = \sqrt{-1}\). So, \(\sqrt{-12} = \sqrt{12} \times i\).
Simplify \(\sqrt{12}\) by factoring it into \(\sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}\). Therefore, \(\sqrt{-12} = 2\sqrt{3}i\).
Substitute back into the original expression: \(\frac{-5 - 2\sqrt{3}i}{48}\). Now, separate the real and imaginary parts by dividing both terms in the numerator by 48.
Write the expression in standard form \(a + bi\) as \(\frac{-5}{48} - \frac{2\sqrt{3}}{48}i\). Simplify the fractions if possible to complete the standard form.
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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. The standard form requires separating real and imaginary components clearly, which is essential when simplifying expressions involving square roots of negative numbers.
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Complex Numbers In Polar Form
Simplifying Square Roots of Negative Numbers
The square root of a negative number involves imaginary units, where √(-x) = i√x. Recognizing and converting these roots into terms involving i is crucial for handling expressions like √(-12), enabling further algebraic manipulation.
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Operations with Complex Numbers
Performing addition, subtraction, multiplication, or division with complex numbers requires combining like terms and applying algebraic rules, including distributing and rationalizing denominators. Mastery of these operations ensures the expression can be simplified correctly into standard form.
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4:22Dividing Complex Numbers
Related Practice
Textbook Question
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. x² + y² = 9
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Textbook Question
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [√3 (cos (5π/18) + i sin (5π/18))]⁶
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Evaluate x² − 2x + 2 for x = 1 + i.
Textbook Question
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Line: Passes through (−2,4) and (1,7)
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