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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
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Tutti i libri di testoBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.2.45
Capitolo 5, Problema 5.2.45

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 = 20(cos 75° + i sin 75°)
z₂ = 4(cos 25° + i sin 25°)

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1
Recall that when dividing two complex numbers in polar form, \(z_1 = r_1(\cos \theta_1 + i \sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i \sin \theta_2)\), the quotient is given by: \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right)\)
Identify the magnitudes and arguments of the given complex numbers: \(r_1 = 20\), \(\theta_1 = 75^\circ\) \(r_2 = 4\), \(\theta_2 = 25^\circ\)
Calculate the magnitude of the quotient by dividing the magnitudes: \(r = \frac{r_1}{r_2} = \frac{20}{4}\)
Calculate the argument of the quotient by subtracting the angles: \(\theta = \theta_1 - \theta_2 = 75^\circ - 25^\circ\)
Write the quotient in polar form using the results from steps 3 and 4: \(\frac{z_1}{z_2} = r \left( \cos \theta + i \sin \theta \right)\), where \(r\) and \(\theta\) are the values found above. Make sure the argument \(\theta\) is expressed between \(0^\circ\) and \(360^\circ\) if necessary.

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Complex Numbers in Polar Form

Complex numbers can be expressed 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/r₂)(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)). This method avoids converting back to rectangular form and makes calculations more straightforward.
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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 standard, positive angle measurement.
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Dividing Complex Numbers
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