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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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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 EquationsProblema 5.1.61
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.1.61Capitolo 5, Problema 5.1.61
Evaluate x²+19 / 2−x for x = 3i.
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Identify the given expression: \(\frac{x^2 + 19}{2 - x}\) and the value of \(x = 3i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
Substitute \(x = 3i\) into the numerator: calculate \(x^2 + 19\) by first finding \(x^2 = (3i)^2\) and then adding 19.
Substitute \(x = 3i\) into the denominator: calculate \(2 - x = 2 - 3i\).
Simplify the numerator using the fact that \(i^2 = -1\), so \((3i)^2 = 9i^2 = 9(-1) = -9\), then add 19 to get the numerator value.
Write the expression as a complex fraction with the simplified numerator and denominator, and if needed, multiply numerator and denominator by the conjugate of the denominator to simplify the expression further.
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Complex Numbers
Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to substitute and manipulate complex numbers is essential when evaluating expressions involving imaginary values.
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Dividing Complex Numbers
Substitution in Algebraic Expressions
Substitution involves replacing a variable in an expression with a given value. When the value is complex, careful algebraic manipulation is required to correctly simplify the expression, especially when dealing with powers and denominators.
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6:36Simplifying Trig Expressions
Simplification of Rational Expressions
Simplifying rational expressions involves performing algebraic operations such as expanding, factoring, and reducing fractions. When the numerator and denominator contain complex numbers, simplification must consider the properties of complex arithmetic to reach the final simplified form.
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2:58Rationalizing Denominators
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In Exercises 37–52, perform the indicated operations and write the result in standard form.
( −6 − √(−12)) / 48
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In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t
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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 = 4 + 2 cos t, y = 3 + 5 sin t; t = π/2
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Evaluate x² − 2x + 2 for x = 1 + i.
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In Exercises 1–10, plot each complex number and find its absolute value. z = 3 + 2i
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