Symmetries and Polar Graphs

Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.

r = 1 + 2 sin θ

검증된 단계별 안내

Recall that to identify symmetries of a polar curve given by \(r = f(\theta)\), we check for symmetry about the polar axis (the x-axis), the line \(\theta = \frac{\pi}{2}\) (the y-axis), and the pole (origin).

For symmetry about the polar axis, replace \(\theta\) by \(-\theta\) and see if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 1 + 2 \sin(-\theta)\) simplifies to the original \(r\).

For symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) by \(\pi - \theta\) and check if \(r = 1 + 2 \sin(\pi - \theta)\) simplifies to the original \(r\).

For symmetry about the pole (origin), replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\), and check if \(-r = 1 + 2 \sin(\theta + \pi)\) can be rearranged to the original equation.

After determining the symmetries, sketch the curve by plotting points for various values of \(\theta\) between \(0\) and \(2\pi\), using the equation \(r = 1 + 2 \sin \theta\), and then reflect the curve according to the symmetries found.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polar Coordinates and Graphing

Polar coordinates represent points using a radius r and an angle θ, differing from Cartesian coordinates. Understanding how to plot points given r(θ) is essential for sketching curves like r = 1 + 2 sin θ, where r changes with θ from 0 to 2π.

Symmetry in Polar Graphs

Symmetry in polar graphs can occur about the polar axis, the line θ = π/2, or the pole (origin). Testing the function for transformations like replacing θ with -θ, π - θ, or θ + π helps identify these symmetries, which simplifies sketching and understanding the curve.

Trigonometric Functions in Polar Equations

Trigonometric functions like sine and cosine influence the shape and symmetry of polar graphs. For r = 1 + 2 sin θ, the sine term affects the radius depending on θ, creating characteristic shapes such as limacons, cardioids, or loops, which are important to recognize.

Symmetries and Polar GraphsIdentify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.r = 1 + 2 sin θ