교과서 질문
In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex sixth roots of 64
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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 71
In Exercises 59–74, convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation.
r = 6 cos θ + 4 sin θ
검증된 단계별 안내1
Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\).
Start with the given polar equation: \(r = 6 \cos \theta + 4 \sin \theta\).
Multiply both sides of the equation by \(r\) to eliminate the \(r\) on the left side: \(r \cdot r = r (6 \cos \theta + 4 \sin \theta)\), which gives \(r^2 = 6r \cos \theta + 4r \sin \theta\).
Substitute the rectangular coordinate equivalents: \(r^2 = x^2 + y^2\), \(r \cos \theta = x\), and \(r \sin \theta = y\). This transforms the equation into \(x^2 + y^2 = 6x + 4y\).
Rearrange the equation to standard form by bringing all terms to one side: \(x^2 - 6x + y^2 - 4y = 0\). Then complete the square for both \(x\) and \(y\) terms to express the equation in the form of a circle.
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 to Rectangular Coordinate Conversion
Polar coordinates (r, θ) relate to rectangular coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Converting polar equations to rectangular form involves substituting these expressions to rewrite the equation solely in terms of x and y.
추천 영상:
06:17
Convert Points from Polar to Rectangular
Trigonometric Identities and Relationships
Understanding the relationships between sine, cosine, and their geometric interpretations is essential. For example, recognizing that cos θ = x/r and sin θ = y/r helps in manipulating and simplifying equations during conversion from polar to rectangular form.
추천 영상:
5:32Fundamental Trigonometric Identities
Graphing in the Rectangular Coordinate System
Once the equation is converted to rectangular form, graphing involves plotting points (x, y) on the Cartesian plane. Familiarity with the shapes of conic sections and lines helps interpret the resulting equation and sketch its graph accurately.
추천 영상:
5:10Introduction to Graphs & the Coordinate System
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