교과서 질문
In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)
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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 17
In Exercises 13–34, test for symmetry and then graph each polar equation.r = 2 + 2 cos θ
검증된 단계별 안내1
Identify the type of symmetry to test: polar equations can have symmetry with respect to the polar axis (x-axis), the line θ = π/2 (y-axis), or the pole (origin).
Test for symmetry with respect to the polar axis by replacing θ with -θ in the equation: r = 2 + 2 \(\cos\)(-θ). Since \(\cos\)(-θ) = \(\cos\)(θ), the equation remains unchanged, indicating symmetry with respect to the polar axis.
Test for symmetry with respect to the line θ = π/2 by replacing θ with π - θ: r = 2 + 2 \(\cos\)(π - θ). Since \(\cos\)(π - θ) = -\(\cos\)(θ), the equation changes, indicating no symmetry with respect to the line θ = π/2.
Test for symmetry with respect to the pole by replacing r with -r: -r = 2 + 2 \(\cos\)(θ). This does not simplify to the original equation, indicating no symmetry with respect to the pole.
Graph the equation by plotting points for various values of θ, using the symmetry with respect to the polar axis to simplify the process. The graph will be a limaçon with an inner loop.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Polar Coordinates
Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In polar equations, 'r' denotes the radius (distance from the origin), and 'θ' represents the angle. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.
추천 영상:
05:32
Intro to Polar Coordinates
Symmetry in Polar Graphs
Symmetry in polar graphs can be tested by substituting specific values for θ. A graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and it is symmetric about the line θ = π/2 if replacing θ with π - θ gives the same result. Recognizing these symmetries helps in sketching the graph accurately.
추천 영상:
3:19Cardioids
Graphing Polar Equations
Graphing polar equations involves plotting points based on the values of 'r' for various angles 'θ'. The shape of the graph can vary significantly depending on the equation's form. For the equation r = 2 + 2 cos θ, understanding how to evaluate 'r' at key angles (like 0, π/2, π, and 3π/2) is crucial for accurately depicting the graph.
추천 영상:
3:47Introduction to Common Polar Equations
관련 실천
교과서 질문
In Exercises 9–20, find each product and write the result in standard form.
(3 + 5i)(3 − 5i)
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교과서 질문
In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)
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교과서 질문
In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)
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교과서 질문
In Exercises 19–21, find the product of the complex numbers. Leave answers in polar form.
z₁ = 3(cos 40°+i sin 40°)
z₂ = 5(cos 70°+i sin 70°)
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교과서 질문
In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−1, π)
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