교과서 질문
In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 7π/4 + i sin 7π/4)
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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 31
In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)
검증된 단계별 안내1
Understand that the point is given in polar coordinates as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
Recall that changing the angle \(\theta\) by adding or subtracting full rotations of \(2\pi\) radians (i.e., \(\theta + 2k\pi\), where \(k\) is an integer) does not change the location of the point because angles are periodic with period \(2\pi\).
Note that changing the radius \(r\) to its negative value \(-r\) and adding \(\pi\) to the angle \(\theta\) (i.e., \((-r, \theta + \pi)\)) also represents the same point, because moving in the opposite direction by \(\pi\) radians with a negative radius points to the same location.
Apply these principles to the given points: for \(( -5, -\frac{\pi}{4} )\), consider if changing the angle by \(2\pi\) or adding \(\pi\) to the angle and negating the radius results in the same point; similarly, analyze \(( -5, \frac{7\pi}{4} )\) under these transformations.
Summarize which representations keep the point's location unchanged by verifying if the transformed coordinates correspond to the same position in the plane using the above rules.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
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Polar Coordinates and Their Representation
Polar coordinates represent points in the plane using a radius and an angle, written as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how points are plotted in polar form is essential to analyze transformations that affect their location.
추천 영상:
05:32
Intro to Polar Coordinates
Equivalent Polar Coordinates
A single point in polar coordinates can have multiple representations due to periodicity of angles and sign changes in radius. For example, (r, θ) is equivalent to (−r, θ + π) and (r, θ + 2πk) for any integer k. Recognizing these equivalences helps identify which transformations preserve the point's location.
추천 영상:
05:32Intro to Polar Coordinates
Effect of Angle and Radius Transformations on Point Location
Changing the angle by adding multiples of 2π or adjusting the radius sign with a corresponding angle shift can produce the same point. However, other transformations may move the point. Understanding how these changes affect the point's position is key to selecting representations that do not alter its location.
추천 영상:
4:25Graphs of Shifted and Reflected Functions
관련 실천
교과서 질문
In Exercises 30–31, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 125(cos 165° + i sin 165°)
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교과서 질문
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교과서 질문
In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of 8i
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교과서 질문
In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)
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교과서 질문
In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)
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