Domanda del libro di testo
In Exercises 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.
P₁ = (-3, 0), P₂ = (-2, -2)
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 4, Problema 26
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³
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Identify the complex number in polar form: \(4(\cos 50^\circ + i \sin 50^\circ)\), where the modulus \(r = 4\) and the argument \(\theta = 50^\circ\).
Recall DeMoivre's Theorem, which states that for a complex number in polar form, \((r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta)\).
Apply DeMoivre's Theorem with \(n = 3\): compute the new modulus as \(r^3 = 4^3\) and the new argument as \(3 \times 50^\circ\).
Write the result in polar form: \(4^3 (\cos 150^\circ + i \sin 150^\circ)\).
Convert the polar form back to rectangular form by calculating \(4^3 \cos 150^\circ\) for the real part and \(4^3 \sin 150^\circ\) for the imaginary part.
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DeMoivre's Theorem
DeMoivre's Theorem states that for a complex number in polar form, (r(cos θ + i sin θ))^n = r^n (cos nθ + i sin nθ). It allows raising complex numbers to integer powers by multiplying the angle and raising the magnitude to the power.
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03:41
Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)
Polar and Rectangular Forms of Complex Numbers
Complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude and θ the argument, or in rectangular form as a + bi. Converting between these forms is essential for interpreting and simplifying results.
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03:58Converting Complex Numbers from Polar to Rectangular Form
Conversion from Polar to Rectangular Form
After applying DeMoivre's Theorem, the result is in polar form. To write the answer in rectangular form, use a = r cos θ and b = r sin θ to find the real and imaginary parts, respectively, expressing the complex number as a + bi.
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03:58Converting Complex Numbers from Polar to Rectangular Form
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Domanda del libro di testo
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