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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

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

Problema 3

Capitolo 5, Problema 3

In Exercises 1–3, perform the indicated operations and write the result in standard form. _ _ 2√−49 + 3√−64

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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: $2\sqrt{-49} = 2\sqrt{49} \times i$ and $3\sqrt{-64} = 3\sqrt{64} \times i$.

Calculate the square roots of the positive numbers: $\sqrt{49} = 7$ and $\sqrt{64} = 8$.

Substitute these values back into the expression: $2 \times 7 \times i + 3 \times 8 \times i$.

Combine like terms by factoring out $i$: $(2 \times 7 + 3 \times 8) i$, which simplifies to $(14 + 24) i$. This is the expression in standard form $a + bi$, where $a$ is the real part and $b$ is the coefficient of the imaginary part.

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Imaginary Numbers and the Imaginary Unit i

Imaginary numbers arise when taking the square root of negative numbers, which is not defined in the real number system. The imaginary unit i is defined as √(-1), allowing us to express roots of negative numbers as multiples of i, such as √(-49) = 7i.

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Imaginary Roots with the Square Root Property

Simplifying Radicals Involving Negative Numbers

To simplify radicals with negative radicands, factor out the negative sign as i² or i, then simplify the positive part. For example, √(-64) = √(64) × √(-1) = 8i. This process helps convert complex radicals into a standard form involving real and imaginary parts.

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Dividing Complex Numbers

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. After performing operations on imaginary numbers, results should be expressed in this form to clearly separate the real and imaginary components.

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04:47

Complex Numbers In Polar Form

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In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, 5π/4)

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_ Write −√3 + i in polar form.

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