Questions: Find the symmetries of the graph of r=4(1-sin theta).

Solution Steps

To determine the symmetries of the polar equation \( r = 4(1 - \sin \theta) \), we need to check for symmetry with respect to the x-axis, y-axis, and the origin. For x-axis symmetry, replace \(\theta\) with \(-\theta\) and check if the equation remains unchanged. For y-axis symmetry, replace \(\theta\) with \(\pi - \theta\). For origin symmetry, replace \((r, \theta)\) with \((-r, \theta + \pi)\).

To determine if the graph of \( r = 4(1 - \sin \theta) \) is symmetrical about the x-axis, we replace \( \theta \) with \( -\theta \): \[ r_x = 4(1 - \sin(-\theta)) = 4(1 + \sin \theta) = 4 + 4\sin \theta \] Comparing \( r \) and \( r_x \): \[ r = 4 - 4\sin \theta \quad \text{and} \quad r_x = 4 + 4\sin \theta \] Since \( r \neq r_x \), the graph is not symmetrical about the x-axis.

Next, we check for symmetry about the y-axis by replacing \( \theta \) with \( \pi - \theta \): \[ r_y = 4(1 - \sin(\pi - \theta)) = 4(1 - \sin \theta) \] Comparing \( r \) and \( r_y \): \[ r = 4 - 4\sin \theta \quad \text{and} \quad r_y = 4 - 4\sin \theta \] Since \( r = r_y \), the graph is symmetrical about the y-axis.

Finally, we check for symmetry about the origin by replacing \( (r, \theta) \) with \( (-r, \theta + \pi) \): \[ r_{\text{origin}} = -4(1 - \sin(\theta + \pi)) = -4(1 + \sin \theta) = -4 - 4\sin \theta \] Comparing \( r \) and \( r_{\text{origin}} \): \[ r = 4 - 4\sin \theta \quad \text{and} \quad r_{\text{origin}} = -4 - 4\sin \theta \] Since \( r \neq r_{\text{origin}} \), the graph is not symmetrical about the origin.

Final Answer