Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

검증된 단계별 안내

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

For symmetry about the polar axis (the horizontal axis), replace \(\theta\) by \(-\theta\) and check if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 4 - \sin\left(\frac{-\theta}{2}\right)\) simplifies to the original \(r\).

For symmetry about the line \(\theta = \frac{\pi}{2}\) (vertical axis), replace \(\theta\) by \(\pi - \theta\) and check if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 4 - \sin\left(\frac{\pi - \theta}{2}\right)\) simplifies to the original \(r\).

For symmetry about the pole (origin), replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\), and check if the equation holds. That is, check if \(-r = 4 - \sin\left(\frac{\theta + \pi}{2}\right)\) can be rearranged to the original equation.

Analyze the results from these substitutions to conclude which symmetries the curve has based on whether the equation remains equivalent after each transformation.

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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 and an angle, with equations expressing relationships between r (radius) and θ (angle). Understanding how to interpret and manipulate polar equations is essential for analyzing curves defined in this system.

Symmetry in Polar Graphs

Symmetry in polar graphs can be tested by checking if the equation remains unchanged under transformations: θ replaced by -θ (symmetry about the polar axis), θ replaced by π - θ (symmetry about the vertical line θ = π/2), or r replaced by -r with θ replaced by θ + π (symmetry about the pole).

Trigonometric Function Properties

Understanding the properties of sine functions, especially with angle transformations like θ/2, helps determine how the function behaves under angle shifts or reflections. This is crucial for testing symmetry without graphing tools.