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Perform the indicated operations and write the result in standard form. (−2 + √−100)²
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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 9
Test for symmetry with respect to a. the polar axis. b. the line θ = π/2. c. the pole. r = 4 + 3 cos θ
검증된 단계별 안내1
Recall the tests for symmetry in polar coordinates:
- Symmetry about the polar axis (the horizontal axis) means that if \((r, \theta)\) is on the graph, then \((r, -\theta)\) is also on the graph.
- Symmetry about the line \(\theta = \frac{\pi}{2}\) means that if \((r, \theta)\) is on the graph, then \((r, \pi - \theta)\) is also on the graph.
- Symmetry about the pole (origin) means that if \((r, \theta)\) is on the graph, then \((-r, \theta)\) or equivalently \((r, \theta + \pi)\) is also on the graph.
To test symmetry about the polar axis, replace \(\theta\) with \(-\theta\) in the equation and check if the equation remains unchanged. For the given equation:
\(r = 4 + 3 \cos \theta\)
Replace \(\theta\) with \(-\theta\):
\(r = 4 + 3 \cos(-\theta)\)
Use the even property of cosine, which states \(\cos(-\theta) = \cos \theta\), so the equation becomes:
\(r = 4 + 3 \cos \theta\)
Since this is the same as the original equation, the graph is symmetric about the polar axis.
To test symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) with \(\pi - \theta\) in the equation and check if the equation remains unchanged:
\(r = 4 + 3 \cos(\pi - \theta)\)
Use the identity \(\cos(\pi - \theta) = -\cos \theta\), so the equation becomes:
\(r = 4 - 3 \cos \theta\)
Since this is not the same as the original equation, the graph is not symmetric about the line \(\theta = \frac{\pi}{2}\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Symmetry with Respect to the Polar Axis
Symmetry about the polar axis means the graph remains unchanged when θ is replaced by -θ. To test this, substitute -θ into the polar equation and check if the resulting equation is equivalent to the original. This reflects symmetry across the horizontal axis in polar coordinates.
추천 영상:
05:32
Intro to Polar Coordinates
Symmetry with Respect to the Line θ = π/2
Symmetry about the line θ = π/2 occurs if replacing θ by π - θ yields an equivalent equation. This tests whether the graph is mirrored across the vertical line θ = π/2 in the polar plane. It helps identify vertical symmetry in polar graphs.
추천 영상:
2:22Cardioids Example 1
Symmetry with Respect to the Pole
Symmetry about the pole (origin) means the graph is unchanged when (r, θ) is replaced by (-r, θ) or equivalently (r, θ + π). Testing this involves checking if r(θ) = -r(θ + π) or if the equation remains valid under these transformations, indicating origin symmetry.
추천 영상:
2:22Cardioids Example 1
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