Skip to main content
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric Equations문제 23
5장, 문제 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)

검증된 단계별 안내
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: \( \frac{5}{10} \left( \cos \left( \frac{4\pi}{3} - \frac{\pi}{3} \right) + i \sin \left( \frac{4\pi}{3} - \frac{\pi}{3} \right) \right) \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
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.
추천 영상:
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.
추천 영상:
5:32
Fundamental Trigonometric Identities
관련 실천
교과서 질문

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
교과서 질문

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

702
views
교과서 질문

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
교과서 질문

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

670
views
교과서 질문

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


3 / 4+i

495
views
교과서 질문

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

643
views