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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 5.2.45
Chapter 5, Problem 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°)

Verified step by step guidance
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
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