Skip to main content
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 23
Chapter 5, Problem 23

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.
z₁ = 5 (cos 4π/3 + i sin 4π/3)
z₂ = 10 (cos π/3 + i sin π/3)

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 \( \frac{z_1}{z_2} \) is given by \( \frac{r_1}{r_2} \left( \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2) \right) \).
Identify the magnitudes and angles from the given complex numbers: \( r_1 = 5 \), \( \theta_1 = \frac{4\pi}{3} \), \( r_2 = 10 \), and \( \theta_2 = \frac{\pi}{3} \).
Calculate the quotient of the magnitudes: \( \frac{r_1}{r_2} = \frac{5}{10} \).
Find the difference of the angles: \( \theta_1 - \theta_2 = \frac{4\pi}{3} - \frac{\pi}{3} \).
Write the quotient \( \frac{z_1}{z_2} \) in polar form using the results from steps 3 and 4: \( \frac{5}{10} \left( \cos \left( \frac{4\pi}{3} - \frac{\pi}{3} \right) + i \sin \left( \frac{4\pi}{3} - \frac{\pi}{3} \right) \right) \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Polar Form of Complex Numbers

A complex number 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 is useful for multiplication and division because it separates the magnitude and angle components.
Recommended video:
04:47
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 angles: (r₁/r₂) [cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)]. This simplifies the operation compared to Cartesian form.
Recommended video:
04:47
Complex Numbers In Polar Form

Trigonometric Identities for Cosine and Sine

Understanding the values of cosine and sine at standard angles (like π/3 and 4π/3) helps in simplifying the final expression. These identities allow you to interpret or convert the polar form results accurately.
Recommended video:
5:32
Fundamental Trigonometric Identities
Related Practice
Textbook Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = t, y = 2t

753
views
Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ

702
views
Textbook Question

In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3 + 4i

528
views
Textbook Question

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

670
views
Textbook Question

In Exercises 21–28, divide and express the result in standard form.


3 / 4+i

495
views
Textbook Question

In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i

643
views