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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.2.49
Capitolo 5, Problema 5.2.49

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°

Guida verificata passo dopo passo
1
Recognize that the complex numbers are given in polar form using Euler's formula: \(z = \cos \theta + i \sin \theta\) corresponds to \(z = r(\cos \theta + i \sin \theta)\) with \(r=1\) here.
Recall the formula for dividing two complex numbers in polar form: if \(z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)\), then their quotient is \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right)\).
Identify the magnitudes and arguments of \(z_1\) and \(z_2\): both have magnitude \(r_1 = r_2 = 1\), and arguments \(\theta_1 = 80^\circ\), \(\theta_2 = 200^\circ\).
Calculate the magnitude of the quotient: \(\frac{r_1}{r_2} = \frac{1}{1} = 1\).
Calculate the argument of the quotient: \(\theta = \theta_1 - \theta_2 = 80^\circ - 200^\circ = -120^\circ\). Since the problem asks for an angle between \(0^\circ\) and \(360^\circ\), add \(360^\circ\) to get \(240^\circ\). Thus, the quotient in polar form is \(1 \left( \cos 240^\circ + i \sin 240^\circ \right)\).

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Complex Numbers in Polar Form

Complex numbers can be represented in polar form as r(cos θ + i sin θ), where r is the magnitude and θ is the argument (angle). This form simplifies multiplication and division by working directly with magnitudes and angles instead of real and imaginary parts.
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Complex Numbers In Polar Form

Division of Complex Numbers in Polar Form

To divide two complex numbers in polar form, divide their magnitudes and subtract the arguments: (r₁∠θ₁) / (r₂∠θ₂) = (r₁/r₂) ∠ (θ₁ - θ₂). This method avoids complicated algebraic manipulation and yields the quotient in polar form.
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Complex Numbers In Polar Form

Argument of a Complex Number and Angle Normalization

The argument of a complex number is the angle it makes with the positive real axis. When expressing the argument, it is often normalized to lie within 0° to 360° by adding or subtracting full rotations (360°) to ensure a positive angle measurement.
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Dividing Complex Numbers
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