교과서 질문
In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1
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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 33
In Exercises 13–34, test for symmetry and then graph each polar equation. r cos θ = −3
검증된 단계별 안내1
Rewrite the given polar equation \(r \cos \theta = -3\) in a more recognizable form. Recall that \(r \cos \theta\) represents the \(x\)-coordinate in Cartesian coordinates, so the equation can be rewritten as \(x = -3\).
Analyze the symmetry of the equation with respect to the polar axes: the polar axis (the positive \(x\)-axis), the line \(\theta = \frac{\pi}{2}\) (the vertical axis), and the pole (origin).
Test for symmetry about the polar axis by replacing \(\theta\) with \(-\theta\) in the original equation and checking if the equation remains unchanged.
Test for symmetry about the line \(\theta = \frac{\pi}{2}\) by replacing \(\theta\) with \(\pi - \theta\) and checking if the equation remains unchanged.
Test for symmetry about the pole by replacing \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\) and checking if the equation remains unchanged. Then, sketch the graph of the line \(x = -3\) in the Cartesian plane, which corresponds to a vertical line 3 units to the left of the origin.
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Polar Coordinates and Equations
Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express relationships between r and θ, describing curves in the plane. Understanding how to interpret and manipulate these equations is essential for graphing and analyzing their properties.
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. Identifying symmetry helps simplify graphing and understanding the curve's shape.
추천 영상:
3:19Cardioids
Conversion Between Polar and Cartesian Coordinates
Converting polar equations to Cartesian form using x = r cos θ and y = r sin θ can simplify analysis. For example, r cos θ = x, so the given equation can be rewritten in Cartesian coordinates to identify the curve type. This conversion aids in graphing and understanding the geometric nature of the equation.
추천 영상:
05:32Intro to Polar Coordinates
관련 실천
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In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)
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교과서 질문
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x = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π
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