Skip to main content
Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric Equations문제 21
5장, 문제 21

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ

검증된 단계별 안내
1
Recall the three common tests for symmetry in polar coordinates: 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 \( r = 1 + 2 \cos \theta \) and check if the equation remains unchanged.
To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged.
To test symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged.
After determining the symmetries, plot points for various values of \( \theta \) (for example, \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), etc.) by calculating \( r \) and then sketch the graph accordingly.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
12m
도움이 되었나요?

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 (x, y). Understanding how to interpret and plot polar equations like r = 1 + 2 cos θ 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 equation remains unchanged under transformations: θ → -θ (symmetry about the polar axis), θ → π - θ (symmetry about the line θ = π/2), and r → -r with θ → θ + π (symmetry about the pole). This helps simplify graphing by identifying mirrored parts.
추천 영상:
3:19
Cardioids

Graphing Polar Equations Involving Cosine

Polar equations with cosine terms, such as r = 1 + 2 cos θ, often produce limaçon shapes. Recognizing how the cosine function affects the radius for different angles allows for accurate plotting and understanding of the curve's features like loops or dimpled shapes.
추천 영상:
3:47
Introduction to Common Polar Equations
관련 실천
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 = 2t

753
views
교과서 질문

In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3

434
views
교과서 질문

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

670
views
교과서 질문

In Exercises 9–20, use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = 2t, y = |t − 1|; −∞ < t < ∞

756
views
교과서 질문

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 + cos θ

704
views
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.


(5, π/6)

742
views