Textbook Question
In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.
z₁ = cos 80° + i sin 80°
z₂ = cos 200° + i sin 200°
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.27
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.27Chapter 5, Problem 5.27
In Exercises 21–28, divide and express the result in standard form.
2+3i / 2+i
Verified step by step guidance1
Identify the given complex division problem: \(\frac{2+3i}{2+i}\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \(2+i\) is \(2 - i\).
Multiply numerator and denominator by the conjugate: \(\frac{2+3i}{2+i} \times \frac{2 - i}{2 - i} = \frac{(2+3i)(2 - i)}{(2+i)(2 - i)}\).
Expand both numerator and denominator using the distributive property (FOIL method):
- Numerator: \((2)(2) + (2)(-i) + (3i)(2) + (3i)(-i)\)
- Denominator: \((2)(2) + (2)(-i) + (i)(2) + (i)(-i)\).
Simplify the expressions by combining like terms and using \(i^2 = -1\), then write the result in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Complex Number Division
Dividing complex numbers involves expressing the quotient in a form that separates real and imaginary parts. This is typically done by multiplying numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.
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Dividing Complex Numbers
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, specifically a^2 + b^2, which helps simplify division by removing the imaginary component from the denominator.
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5:33Complex Conjugates
Standard Form of a Complex Number
The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Expressing results in this form makes it easier to interpret and use complex numbers in further calculations.
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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.
x² = 6y
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Textbook Question
In Exercises 37–52, perform the indicated operations and write the result in standard form.
√(−8) (√(−3) − √5 )
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Textbook Question
In Exercises 29–36, simplify and write the result in standard form.
√3² − 4 ⋅ 2 ⋅ 5
Textbook Question
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 40° + i sin 40°)]³
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Textbook Question
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Ellipse: Center: (−2,3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center
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