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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 37
Capitolo 5, Problema 37

In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ √−64 − √−25

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Recognize that the square roots of negative numbers involve imaginary numbers. Recall that \(\sqrt{-a} = \sqrt{a} \times i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
Rewrite each term using the imaginary unit: \(\sqrt{-64} = \sqrt{64} \times i\) and \(\sqrt{-25} = \sqrt{25} \times i\).
Calculate the square roots of the positive numbers: \(\sqrt{64} = 8\) and \(\sqrt{25} = 5\).
Substitute these values back into the expression: \(8i - 5i\).
Combine like terms (both are imaginary terms) to write the result in standard form $a + bi$, where \(a\) and \(b\) are real numbers.

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Imaginary Numbers and the Imaginary Unit i

Imaginary numbers arise when taking the square root of negative numbers. The imaginary unit i is defined as √−1, allowing us to express roots of negative numbers as multiples of i, such as √−64 = 8i.
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Imaginary Roots with the Square Root Property

Simplifying Square Roots of Negative Numbers

To simplify the square root of a negative number, separate it into the square root of the positive part and the imaginary unit i. For example, √−25 = √25 × √−1 = 5i, which helps in performing arithmetic operations.
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Percorso guidato
2:20
Imaginary 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. After performing operations with imaginary numbers, express the result in this form to clearly separate the real and imaginary parts.
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04:47
Complex Numbers In Polar Form
Pratica correlata
Domanda del libro di testo

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 = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

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Domanda del libro di testo

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (7.4, 2.5)

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Domanda del libro di testo

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (−4, π/2)

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Domanda del libro di testo

In Exercises 37–44, find the product of the complex numbers. Leave answers in polar form. z₁ = 6(cos 20° + i sin 20°) z₂ = 5(cos 50° + i sin 50°)

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Domanda del libro di testo

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 = 5 sec t, y = 3 tan t

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Domanda del libro di testo

In Exercises 29–36, simplify and write the result in standard form. ____________ √1² − 4 ⋅ 0.5 ⋅ 5

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