Domanda del libro di testo
In Exercises 27–32, select the representations that do not change the location of the given point. (−2, 7π/6) (−2, −5π/6)
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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 EquationsProblema 29
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 29Capitolo 5, Problema 29
In Exercises 29–36, simplify and write the result in standard form. ___ √−49
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Recognize that the expression involves the square root of a negative number, specifically \(\sqrt{-49}\). This indicates the use of imaginary numbers since the square root of a negative number is not defined in the set of real numbers.
Recall the definition of the imaginary unit \(i\), where \(i = \sqrt{-1}\). This allows us to rewrite the square root of a negative number in terms of \(i\).
Express \(\sqrt{-49}\) as \(\sqrt{49 \times -1}\), which can be separated into \(\sqrt{49} \times \sqrt{-1}\) using the property of square roots.
Calculate \(\sqrt{49}\), which is a positive real number, and replace \(\sqrt{-1}\) with \(i\). So, \(\sqrt{-49} = \sqrt{49} \times i\).
Write the simplified expression in standard form for complex numbers, which is $a + bi$. Since there is no real part here, the expression will be purely imaginary.
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Imaginary Numbers
Imaginary numbers extend the real number system by including the square root of negative one, denoted as i. Since the square root of a negative number is not real, it is expressed using i, where i² = -1. For example, √-49 can be written as 7i.
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3:31
Introduction to Complex Numbers
Simplifying Square Roots of Negative Numbers
To simplify the square root of a negative number, separate it into the square root of the negative sign and the square root of the positive number. For instance, √-49 = √-1 × √49 = i × 7 = 7i. This process helps convert complex roots into standard imaginary form.
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2:20Imaginary Roots with the Square Root Property
Standard Form of Complex Numbers
The standard form of a complex number is a + bi, where a and b are real numbers and i is the imaginary unit. When simplifying expressions like √-49, the result is purely imaginary (0 + 7i). Writing answers in this form clarifies the real and imaginary parts.
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04:47Complex Numbers In Polar Form
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Domanda del libro di testo
In Exercises 29–36, simplify and write the result in standard form.
√−196
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Domanda del libro di testo
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (−2 − 2i)⁵
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Domanda del libro di testo
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. _ (1−i√3)²
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In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = 1 + 3 cos t, y = 2 + 3 sin t; 0 ≤ t < 2π
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Domanda del libro di testo
In Exercises 27–32, select the representations that do not change the location of the given point. (2, − 3π/4) (2, − 7π/4)
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