교과서 질문
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
x² + y² = 16
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
모든 교과서
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric Equations문제 5.3.56
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric Equations문제 5.3.565장, 문제 5.3.56
In Exercises 49–58, convert each rectangular equation to a polar equation that expresses r in terms of θ.
x² + (y + 3)² = 9
검증된 단계별 안내1
Recall the relationships between rectangular coordinates (x, y) and polar coordinates (r, \(\theta\)): \(x = r \cos\theta\) and \(y = r \sin\theta\).
Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the given equation \(x^2 + (y + 3)^2 = 9\) to rewrite it in terms of \(r\) and \(\theta\).
Expand the expression: \(x^2\) becomes \((r \cos\theta)^2 = r^2 \cos^2\theta\), and \((y + 3)^2\) becomes \((r \sin\theta + 3)^2\).
Write the equation as \(r^2 \cos^2\theta + (r \sin\theta + 3)^2 = 9\) and expand the squared term: \((r \sin\theta + 3)^2 = r^2 \sin^2\theta + 6r \sin\theta + 9\).
Combine like terms and simplify the equation to isolate \(r\) in terms of \(\theta\). Use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\) to simplify \(r^2\) terms.
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주요 개념
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Rectangular and Polar Coordinate Systems
Rectangular coordinates represent points using (x, y) on a plane, while polar coordinates use (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how these systems relate is essential for converting equations between them.
추천 영상:
05:32
Intro to Polar Coordinates
Conversion Formulas Between Rectangular and Polar Coordinates
The key formulas are x = r cos θ and y = r sin θ, which allow substitution of rectangular variables with polar expressions. These formulas enable rewriting equations from rectangular form into polar form by expressing x and y in terms of r and θ.
추천 영상:
06:17Convert Points from Polar to Rectangular
Equation of a Circle in Polar Coordinates
A circle centered at (h, k) with radius a in rectangular form is (x - h)² + (y - k)² = a². When converting to polar, substitute x and y with r cos θ and r sin θ, then simplify to express r as a function of θ, which may involve algebraic manipulation to isolate r.
추천 영상:
05:32Intro to Polar Coordinates
관련 실천
교과서 질문
In Exercises 53–58, perform the indicated operation(s) and write the result in standard form. (2 + i)² − (3 − i)²
760
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교과서 질문
In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, π)
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교과서 질문
In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (5, 0)
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교과서 질문
In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/10 + i sin π/10)]⁵
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교과서 질문
In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.
Line: Passes through (−2,4) and (1,7)
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