교과서 질문
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)
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
5장, 문제 23
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ
검증된 단계별 안내1
Recall that to test for symmetry in polar equations, we check three types of symmetry: symmetry about the polar axis (the horizontal axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (the vertical axis), and symmetry about the pole (the origin).
To test symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and see if the equation remains unchanged. For the given equation \( r = 2 - 3 \sin \theta \), substitute \( \theta \) with \( -\theta \) to get \( r = 2 - 3 \sin(-\theta) \).
Recall that \( \sin(-\theta) = -\sin \theta \), so the equation becomes \( r = 2 + 3 \sin \theta \). Since this is not the same as the original equation, the graph is not symmetric about the polar axis.
To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \theta \) with \( \pi - \theta \) in the original equation: \( r = 2 - 3 \sin(\pi - \theta) \).
Use the identity \( \sin(\pi - \theta) = \sin \theta \), so the equation becomes \( r = 2 - 3 \sin \theta \), which is the same as the original. Therefore, the graph is symmetric about the line \( \theta = \frac{\pi}{2} \).
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Polar Coordinates and Equations
Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing polar curves.
추천 영상:
05:32
Intro to Polar Coordinates
Symmetry Tests in Polar Graphs
Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). These tests involve substituting θ with -θ, π - θ, or replacing r with -r to check if the equation remains unchanged, helping to identify symmetrical properties of the curve.
추천 영상:
3:19Cardioids
Graphing Polar Equations Involving Sine
Polar equations with sine functions, like r = 2 − 3 sin θ, often produce limaçon or cardioid shapes. Recognizing the role of sine in shifting and shaping the curve aids in sketching the graph accurately by evaluating r at key angles and understanding the curve's behavior.
추천 영상:
3:47Introduction to Common Polar Equations
관련 실천
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
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교과서 질문
In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i
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교과서 질문
In Exercises 21–28, divide and express the result in standard form.
3 / 4+i
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교과서 질문
In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i
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교과서 질문
In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)
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