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To determine whether the given polar equation [tex]\( r = \frac{7}{1 + \sin \theta} \)[/tex] exhibits symmetry with respect to the line [tex]\(\theta = \pi / 2\)[/tex], the polar axis, and the pole, we should analyze each type of symmetry individually.
### Symmetry with respect to [tex]\(\theta = \pi / 2\)[/tex]:
To test symmetry with respect to [tex]\(\theta = \pi / 2\)[/tex], we substitute [tex]\(\theta\)[/tex] with [tex]\(\pi - \theta\)[/tex] and see if the equation remains unchanged.
Substitute [tex]\(\theta\)[/tex] with [tex]\(\pi - \theta\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(\pi - \theta)} \][/tex]
Using the trigonometric identity [tex]\(\sin(\pi - \theta) = \sin(\theta)\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(\theta)} \][/tex]
Since this is identical to the original equation, the equation is symmetric with respect to the line [tex]\(\theta = \pi / 2\)[/tex].
### Symmetry with respect to the polar axis:
To test symmetry with respect to the polar axis, we substitute [tex]\(\theta\)[/tex] with [tex]\(-\theta\)[/tex] and see if the equation remains unchanged.
Substitute [tex]\(\theta\)[/tex] with [tex]\(-\theta\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(-\theta)} \][/tex]
Using the trigonometric identity [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex]:
[tex]\[ r = \frac{7}{1 - \sin(\theta)} \][/tex]
Since this is not identical to the original equation, the equation is not symmetric with respect to the polar axis.
### Symmetry with respect to the pole:
To test symmetry with respect to the pole, we replace [tex]\(r\)[/tex] with [tex]\(-r\)[/tex] and see if the equation can be made unchanged by proper manipulation.
Substitute [tex]\(r\)[/tex] with [tex]\(-r\)[/tex]:
[tex]\[ -r = \frac{7}{1 + \sin(\theta)} \][/tex]
Or equivalently,
[tex]\[ r = -\frac{7}{1 + \sin(\theta)} \][/tex]
Since this does not match the original equation (because [tex]\(r\)[/tex] must be positive in the polar coordinate system), the equation is not symmetric with respect to the pole.
### Summary:
- The equation [tex]\(r = \frac{7}{1 + \sin \theta}\)[/tex] is symmetric with respect to the line [tex]\(\theta = \pi / 2\)[/tex].
- The equation is not symmetric with respect to the polar axis.
- The equation is not symmetric with respect to the pole.
So, the correct options are:
- Symmetric with respect to [tex]\(\theta = \pi / 2\)[/tex]: Yes
- Symmetric with respect to the pole: No
- Symmetric with respect to the polar axis: No
### Symmetry with respect to [tex]\(\theta = \pi / 2\)[/tex]:
To test symmetry with respect to [tex]\(\theta = \pi / 2\)[/tex], we substitute [tex]\(\theta\)[/tex] with [tex]\(\pi - \theta\)[/tex] and see if the equation remains unchanged.
Substitute [tex]\(\theta\)[/tex] with [tex]\(\pi - \theta\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(\pi - \theta)} \][/tex]
Using the trigonometric identity [tex]\(\sin(\pi - \theta) = \sin(\theta)\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(\theta)} \][/tex]
Since this is identical to the original equation, the equation is symmetric with respect to the line [tex]\(\theta = \pi / 2\)[/tex].
### Symmetry with respect to the polar axis:
To test symmetry with respect to the polar axis, we substitute [tex]\(\theta\)[/tex] with [tex]\(-\theta\)[/tex] and see if the equation remains unchanged.
Substitute [tex]\(\theta\)[/tex] with [tex]\(-\theta\)[/tex]:
[tex]\[ r = \frac{7}{1 + \sin(-\theta)} \][/tex]
Using the trigonometric identity [tex]\(\sin(-\theta) = -\sin(\theta)\)[/tex]:
[tex]\[ r = \frac{7}{1 - \sin(\theta)} \][/tex]
Since this is not identical to the original equation, the equation is not symmetric with respect to the polar axis.
### Symmetry with respect to the pole:
To test symmetry with respect to the pole, we replace [tex]\(r\)[/tex] with [tex]\(-r\)[/tex] and see if the equation can be made unchanged by proper manipulation.
Substitute [tex]\(r\)[/tex] with [tex]\(-r\)[/tex]:
[tex]\[ -r = \frac{7}{1 + \sin(\theta)} \][/tex]
Or equivalently,
[tex]\[ r = -\frac{7}{1 + \sin(\theta)} \][/tex]
Since this does not match the original equation (because [tex]\(r\)[/tex] must be positive in the polar coordinate system), the equation is not symmetric with respect to the pole.
### Summary:
- The equation [tex]\(r = \frac{7}{1 + \sin \theta}\)[/tex] is symmetric with respect to the line [tex]\(\theta = \pi / 2\)[/tex].
- The equation is not symmetric with respect to the polar axis.
- The equation is not symmetric with respect to the pole.
So, the correct options are:
- Symmetric with respect to [tex]\(\theta = \pi / 2\)[/tex]: Yes
- Symmetric with respect to the pole: No
- Symmetric with respect to the polar axis: No
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