Domanda del libro di testo
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. (1 + i)⁵
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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 5.5.47
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.5.47Capitolo 5, Problema 5.5.47
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
Guida verificata passo dopo passo1
Identify the center of the ellipse as given: \((-2, 3)\).
Determine the lengths of the major and minor axes from the vertices and endpoints: the major axis length is \(5 + 5 = 10\) units, so the semi-major axis \(a = 5\); the minor axis length is \(2 + 2 = 4\) units, so the semi-minor axis \(b = 2\).
Since the vertices are 5 units to the left and right of the center, the major axis is horizontal. This means the parametric equations will be based on \(x\) varying with cosine and \(y\) varying with sine.
Write the parametric equations for the ellipse centered at \((h, k)\) with horizontal major axis:
\(x = h + a \cos(t)\)
\(y = k + b \sin(t)\)
where \(t\) is the parameter varying from \(0\) to \(2\pi\).
Substitute the known values \(h = -2\), \(k = 3\), \(a = 5\), and \(b = 2\) into the parametric equations to get the final set of parametric equations.
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Parametric Equations of Conic Sections
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For conic sections like ellipses, these equations allow a clear representation of the curve by defining x and y in terms of trigonometric functions, simplifying analysis and graphing.
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Percorso guidato
08:02
Parameterizing Equations
Ellipse Geometry and Parameters
An ellipse is defined by its center, vertices, and endpoints of the minor axis. The distance from the center to a vertex is the semi-major axis (a), and the distance to the minor axis endpoint is the semi-minor axis (b). These parameters determine the shape and size of the ellipse.
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Percorso guidato
05:59Eliminating the Parameter
Using Center and Axis Lengths to Form Parametric Equations
To write parametric equations for an ellipse, use the center coordinates as offsets and the semi-major and semi-minor axes as coefficients of cosine and sine functions, respectively. This approach translates geometric information into algebraic form for precise curve description.
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Percorso guidato
08:02Parameterizing Equations
Pratica correlata
Domanda del libro di testo
In Exercises 29–36, simplify and write the result in standard form.
√3² − 4 ⋅ 2 ⋅ 5
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Domanda del libro di testo
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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Domanda del libro di testo
In Exercises 21–28, divide and express the result in standard form.
2+3i / 2+i
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Domanda del libro di testo
In Exercises 1–10, perform the indicated operations and write the result in standard form. (3 − 4i)²
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Domanda del libro di testo
In Exercises 53–56, find two different sets of parametric equations for each rectangular equation. y = x² + 4
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