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 7

Chapter 5, Problem 7

Perform the indicated operations and write the result in standard form. 3+4i / 4−2i

Verified step by step guidance

Identify the given complex division problem: \(\frac{3+4i}{4-2i}\).

To simplify the division of complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4-2i\) is \(4+2i\).

Multiply the numerator: \((3+4i)(4+2i)\) using the distributive property (FOIL method).

Multiply the denominator: \((4-2i)(4+2i)\), which is a difference of squares and simplifies to \(4^2 - (2i)^2\).

After performing the multiplications, separate the real and imaginary parts in the numerator and divide each by the real number obtained in the denominator to write the result in standard form \(a + bi\).

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Key Concepts

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

Complex Numbers and Standard Form

Complex numbers are expressed as a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly in this form, separating real and imaginary components for clarity.

Division of Complex Numbers

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

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part in the denominator, as (a + bi)(a - bi) equals a² + b², a real number, facilitating division.

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Complex Conjugates

Related Practice

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In Exercises 1–10, perform the indicated operations and write the result in standard form. 6 / 5+i

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In Exercises 1–8, add or subtract as indicated and write the result in standard form. 8i − (14 − 9i)

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In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other. x = t and y = t² − 4

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Indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, −135°)

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Perform the indicated operation. Leave answers in polar form. [2(cos 10° + i sin 10°)]⁵

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In Exercises 1–8, parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. x = 2 + 3 cos t, y = 4 + 2 sin t; t = π