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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 39
Capitolo 5, Problema 39

In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ ___ 5√−16 + 3√−81

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Recognize that the expressions involve square roots of negative numbers, which means we are dealing with 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 by separating the negative sign inside the square root: \(5\sqrt{-16} = 5 \times \sqrt{16} \times i\) and \(3\sqrt{-81} = 3 \times \sqrt{81} \times i\).
Calculate the square roots of the positive numbers: \(\sqrt{16} = 4\) and \(\sqrt{81} = 9\), so the expressions become \(5 \times 4 \times i\) and \(3 \times 9 \times i\) respectively.
Multiply the coefficients: \(5 \times 4 = 20\) and \(3 \times 9 = 27\), so the terms are \$20i$ and \$27i$.
Add the two imaginary terms together: \(20i + 27i = (20 + 27)i\), which simplifies to \$47i$. This is the result in standard form, where the real part is 0 and the imaginary part is \(47\).

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Imaginary Numbers and Complex Numbers

Imaginary numbers arise from the square roots of negative numbers, defined using the imaginary unit i, where i² = -1. Complex numbers combine real and imaginary parts in the form a + bi, allowing operations involving roots of negative numbers to be expressed in standard form.
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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 positive part times the square root of -1. For example, √-16 = √16 × √-1 = 4i. This process converts the expression into a form involving imaginary units, facilitating further arithmetic.
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Imaginary Roots with the Square Root Property

Addition of Complex Numbers and Standard Form

Adding complex numbers involves combining their real parts and imaginary parts separately. The standard form of a complex number is a + bi, where a and b are real numbers. Writing results in this form ensures clarity and consistency in representing complex expressions.
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Complex Numbers In Polar Form
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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 = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

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In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−2, 2)

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In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−2 + √−4)²

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