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 80° + i sin 80°)]³
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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 5.5.45
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.5.45Capitolo 5, Problema 5.5.45
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.
Circle: Center: (3,5); Radius: 6
Guida verificata passo dopo passo1
Recall the standard parametric equations for a circle with center \((h, k)\) and radius \(r\):
\[ x = h + r \cos(t) \]
\[ y = k + r \sin(t) \]
Identify the center \((h, k)\) and radius \(r\) from the problem: center \((3, 5)\) and radius \(6\).
Substitute these values into the parametric equations to get:
\[ x = 3 + 6 \cos(t) \]
\[ y = 5 + 6 \sin(t) \]
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Parametric Equations of a Circle
Parametric equations represent a circle by expressing x and y coordinates as functions of a parameter, usually t. For a circle centered at (h, k) with radius r, the equations are x = h + r cos(t) and y = k + r sin(t), where t varies from 0 to 2π.
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08:02
Parameterizing Equations
Understanding the Center and Radius
The center (h, k) of a circle determines its position on the coordinate plane, while the radius r defines its size. These values are essential for constructing the parametric equations, as they shift and scale the standard unit circle equations accordingly.
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06:11Introduction to the Unit Circle
Parameter Range and Interpretation
The parameter t typically represents an angle in radians and varies from 0 to 2π to trace the entire circle once. Understanding this range helps in visualizing the motion along the circle and ensures the parametric equations cover all points on the circle.
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05:59Eliminating the Parameter
Pratica correlata
Domanda del libro di testo
In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. r = 4 csc θ
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Domanda del libro di testo
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°
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Domanda del libro di testo
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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Domanda del libro di testo
In Exercises 1–8, add or subtract as indicated and write the result in standard form. 6 − (−5 + 4i) − (−13 − i)
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Domanda del libro di testo
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ. 3x + y = 7
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