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 21

Chapter 5, Problem 21

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

Verified step by step guidance

Identify the complex number in the denominator: \(3 - i\).

To divide by a complex number, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(3 - i\) is \(3 + i\).

Multiply numerator and denominator by \(3 + i\): \(\frac{2}{3 - i} \times \frac{3 + i}{3 + i} = \frac{2(3 + i)}{(3 - i)(3 + i)}\).

Expand the numerator: \(2(3 + i) = 6 + 2i\).

Expand the denominator using the difference of squares formula: \((3 - i)(3 + i) = 3^2 - (i)^2 = 9 - (-1) = 9 + 1 = 10\).

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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

A complex number is expressed in standard form as a + bi, where a is the real part and b is the imaginary part. Writing results in this form helps clearly separate the real and imaginary components, making it easier to interpret and use in further calculations.

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 converts the division into a simpler form that can be expressed as a standard complex number.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part from the denominator because (a + bi)(a - bi) equals a² + b², a real number. This technique is essential for simplifying complex fractions.

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Complex Conjugates

Related Practice

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 = 2t

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ

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 21–28, divide and express the result in standard form.

3 / 4+i

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Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ

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.

(5, π/6)