교과서 질문
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 20° + i sin 20°)]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 27
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ
검증된 단계별 안내1
Recall the three common symmetry tests for polar equations: symmetry about the polar axis (x-axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (y-axis), and symmetry about the pole (origin).
To test symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and check if the equation remains unchanged. For \( r = 4 \sin 3\theta \), substitute \( -\theta \) to get \( r = 4 \sin(-3\theta) \).
To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \pi - \theta \) into \( r = 4 \sin 3\theta \) to get \( r = 4 \sin 3(\pi - \theta) \).
To test symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged. Substitute these into the equation to get \( -r = 4 \sin 3(\theta + \pi) \).
After determining the symmetries, sketch the graph by plotting points for values of \( \theta \) from 0 to \( 2\pi \), using the equation \( r = 4 \sin 3\theta \). Note that the factor 3 inside the sine function indicates the graph will have multiple petals (specifically, 3 petals if \( n \) is odd in \( r = a \sin n\theta \)).
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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 and an angle (r, θ) instead of Cartesian coordinates. Understanding how to interpret and plot equations like r = 4 sin 3θ is essential for graphing curves in the polar plane.
추천 영상:
05:32
Intro to Polar Coordinates
Symmetry Tests in Polar Graphs
Testing symmetry in polar graphs involves checking if the graph is symmetric about the polar axis, the line θ = π/2, or the pole. This is done by substituting θ with -θ, π - θ, or replacing r with -r, helping to simplify graphing and understand the curve's shape.
추천 영상:
3:19Cardioids
Graphing Rose Curves
Equations of the form r = a sin(nθ) or r = a cos(nθ) produce rose curves with petals. The number of petals depends on n: if n is odd, the curve has n petals; if even, it has 2n petals. Recognizing this helps in sketching the graph of r = 4 sin 3θ accurately.
추천 영상:
3:37Roses
관련 실천
교과서 질문
In Exercises 27–32, select the representations that do not change the location of the given point. (7, 140°) (−7, 320°)
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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 = t − 1
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교과서 질문
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/14 + i sin π/14)]⁷
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교과서 질문
In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)
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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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