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 4

Chapter 5, Problem 4

In Exercises 1–10, perform the indicated operations and write the result in standard form. (3 − 4i)²

Verified step by step guidance

Recall that to square a complex number in the form \((a + bi)\), you use the formula \((a + bi)^2 = a^2 + 2abi + (bi)^2\).

Identify the real part \(a = 3\) and the imaginary part \(b = -4\) from the complex number \((3 - 4i)\).

Apply the formula: calculate \(a^2 = 3^2\), \(2ab i = 2 \times 3 \times (-4) i\), and \((bi)^2 = (-4i)^2\) separately.

Remember that \(i^2 = -1\), so simplify \((bi)^2\) accordingly.

Combine the real parts and the imaginary parts to write the result in standard form \(x + yi\).

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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 in the form a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly as a sum of a real number and an imaginary number, making it easier to interpret and use in further calculations.

Binomial Expansion (Squaring a Complex Number)

Squaring a complex number like (3 − 4i) involves applying the binomial formula (a − b)² = a² − 2ab + b². This requires careful handling of the imaginary unit i, remembering that i² = −1, which affects the sign and value of the terms.

Properties of the Imaginary Unit i

The imaginary unit i is defined such that i² = −1. This property is crucial when simplifying powers of i during operations with complex numbers, as it transforms terms involving i² into real numbers, enabling the expression to be written in standard form.

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Related Practice

Textbook Question

In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. (x − 2)² + y² = 4

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Textbook Question

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

Textbook Question

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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Textbook Question

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 = t² + 3, y = 6 − t³; t = 2

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Textbook Question

In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. ___ ___ 5√(−16) + 3√(−81)

Textbook Question

_ Write −√3 + i in polar form.

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