In Exercises 35–44, test for symmetry and then graph each polar equation. r = cos θ/2

검증된 단계별 안내

Identify the given polar equation: \(r = \cos \frac{\theta}{2}\).

Test for symmetry about the polar axis (the horizontal axis): Replace \(\theta\) with \(-\theta\) and check if the equation remains unchanged. That is, check if \(r(\theta) = r(-\theta)\).

Test for symmetry about the line \(\theta = \frac{\pi}{2}\) (the vertical axis): Replace \(\theta\) with \(\pi - \theta\) and check if the equation remains unchanged. That is, check if \(r(\theta) = r(\pi - \theta)\).

Test for symmetry about the pole (origin): Replace \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\) and check if the equation remains unchanged. That is, check if \(r(\theta) = -r(\theta + \pi)\).

Use the results of the symmetry tests to determine which symmetries the graph has, then plot points for various values of \(\theta\) to sketch the graph of \(r = \cos \frac{\theta}{2}\).

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 Coordinates and Polar Equations

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing.

Symmetry Tests in Polar Graphs

Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). These tests involve substituting -θ, π - θ, or -r into the equation to check if the equation remains unchanged, helping to simplify graphing.

Graphing r = cos(θ/2)

The equation r = cos(θ/2) involves a half-angle, which affects the periodicity and shape of the graph. Recognizing how the cosine function behaves with θ/2 helps predict the number of petals or loops and their orientation in the polar plot.