11.8 SUMMARY

• A point \(P=(x,y)\) has polar coordinates \((r,\theta)\), where \(r\) is the distance to the origin and \(\theta\) is the angle between the positive \(x\)-axis and the segment \(\overline{OP}\), measured in the counterclockwise direction.

  \(x=r\cos\theta\)	\(r=\sqrt{x^2+y^2}\)
  \(y = r\sin\theta\)	\(\tan\theta=\frac{y}{x}\quad (x\neq 0)\)

• The angular coordinate \(\theta\) must be chosen so that \((r,\theta)\) lies in the proper quadrant. If \(r>0\), then \[ \theta = \begin{cases} \tan^{-1} \dfrac{y}{x}& \text{if \(x > 0\)}\\ \tan^{-1} \dfrac{y}{x} + \pi& \text{if \(x < 0\)}\\ \pm \dfrac{\pi}{2}& \text{if \(x = 0\)} \end{cases} \]
• Nonuniqueness: \((r,\theta)\) and \((r,\theta + 2n\pi)\) represent the same point for all integers \(n\). The origin \(O\) has polar coordinates \((0,\theta)\) for any \(\theta\).
• Negative radial coordinates: \((-r,\theta)\) and \((r,\theta+\pi)\) represent the same point.
• Polar equations:

  Curve	Polar equation
  Circle of radius \(R\), center at the origin	\(r = R\)
  Line through origin of slope \(m=\tan\theta_0\)	\(\theta = \theta_0\)
  Line on which \(P_0=(d,\alpha)\) is the point closest to the origin	\(r = d\sec(\theta-\alpha)\)
  Circle of radius \(a\), center at \((a,0)\) \((x-a)^2 + y^2 = a^2\)	\(r = 2a\cos\theta\)
  Circle of radius \(a\), center at \((0,a)\) \(x^2 + (y-a)^2 = a^2\)	\(r = 2a\sin\theta\)