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. [4(cos 15° + i sin 15°)]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 5.3.57
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.3.57Capitolo 5, Problema 5.3.57
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
y² = 6x
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Recall the relationships between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)):
\(x = r \cos{\theta}\)
\(y = r \sin{\theta}\)
Substitute \(x\) and \(y\) in the given equation \(y^2 = 6x\) using the polar forms:
\((r \sin{\theta})^2 = 6 (r \cos{\theta})\)
Simplify the equation:
\(r^2 \sin^2{\theta} = 6r \cos{\theta}\)
Since \(r\) can be zero, consider dividing both sides by \(r\) (assuming \(r \neq 0\)) to isolate \(r\):
\(r \sin^2{\theta} = 6 \cos{\theta}\)
Finally, solve for \(r\) in terms of \(\theta\):
\(r = \frac{6 \cos{\theta}}{\sin^2{\theta}}\)
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Rectangular and Polar Coordinate Systems
Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.
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Percorso guidato
05:32
Intro to Polar Coordinates
Conversion Formulas Between Rectangular and Polar Coordinates
The key formulas are x = r cos θ and y = r sin θ, which allow substitution of rectangular variables with polar expressions. These formulas enable rewriting equations from rectangular form into polar form by expressing x and y in terms of r and θ.
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Percorso guidato
06:17Convert Points from Polar to Rectangular
Algebraic Manipulation to Express r in Terms of θ
After substituting x and y with their polar equivalents, algebraic techniques are used to isolate r on one side of the equation. This step is crucial to express r explicitly as a function of θ, completing the conversion to a polar equation.
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Percorso guidato
6:50Convert Equations from Polar to Rectangular
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