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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

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

Problem 25

Chapter 5, Problem 25

In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i

Verified step by step guidance

Identify the given expression to simplify: \(\frac{8i}{4 - 3i}\).

To express the result in standard form (a + bi), multiply the numerator and denominator by the complex conjugate of the denominator. The conjugate of \(4 - 3i\) is \(4 + 3i\).

Multiply numerator and denominator by \(4 + 3i\): \(\frac{8i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i} = \frac{8i(4 + 3i)}{(4 - 3i)(4 + 3i)}\).

Expand the numerator using distributive property: \(8i \times 4 + 8i \times 3i = 32i + 24i^2\). Remember that \(i^2 = -1\).

Expand the denominator using the difference of squares formula: \((4)^2 - (3i)^2 = 16 - 9i^2\). Again, use \(i^2 = -1\) to simplify.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Number Standard Form

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. Writing complex numbers in this form makes it easier to perform arithmetic operations and interpret their values.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression and allows it to be written in standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, which is useful for rationalizing denominators when dividing complex numbers.

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Complex Conjugates

Related Practice

Textbook Question

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 20° + i sin 20°)]³

Textbook Question

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.

z₁ = 5 (cos 4π/3 + i sin 4π/3)

z₂ = 10 (cos π/3 + i sin π/3)

Textbook Question

In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3 + 4i

Textbook Question

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 = √t, y = t − 1

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Textbook Question

In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)

Textbook Question

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)