Polar coordinates：练习题

Convert the cartesian coordinates \((-\sqrt{3}, 1)\) to polar coordinates \((r, \theta)\), where \(r > 0\) and \(-\pi < \theta \leq \pi\).

A curve has the polar equation \(r = 2\sin \theta\). Find the gradient of the tangent to the curve at the point where \(\theta = \frac{\pi}{6}\).

The region \(R\) lies inside the circle with equation \(r = 3\sin\theta\) and outside the cardioid with equation \(r = 1 + \sin\theta\). Calculate the area of the region \(R\).

A curve is defined by the Cartesian equation \(y = \sqrt{3}x + 2\). Which of the following is the correct polar form of this equation?

Find the total area enclosed by the cardioid with polar equation \(r = a(1 + \cos\theta)\), where \(a\) is a positive constant.

A point has Cartesian coordinates \((-\sqrt{3}, 1)\). Find the polar coordinates \((r, \theta)\) of this point, giving the value of \(\theta\) in terms of \(\pi\) such that \(-\pi < \theta \leq \pi\).

Find the exact area of the region enclosed by the polar curve \(r = 2 + \cos \theta\) for the interval \(0 \le \theta \le \pi\).

A region \(R\) is defined as the area that lies inside the circle with polar equation \(r = 2\) and outside the curve with polar equation \(r = 2(1 - \cos \theta)\). Calculate the exact area of the region \(R\).

A curve C has the polar equation \(r = a(1 + \sin \theta)\) for \(0 \le \theta < 2\pi\), where \(a > 0\).
(a) Sketch the curve C, clearly labeling any intersections with the initial line and the pole.
(b) Show that the total area enclosed by the curve C is \(\frac{3}{2}\pi a^{2}\).
(c) Find the cartesian equation of the tangent to the curve C at the point where \(\theta = \frac{\pi}{6}\).

The curves \( C_1 \) and \( C_2 \) have polar equations \( r = 3 \cos \theta \) and \( r = 1 + \cos \theta \) respectively.
(a) Find the polar coordinates of the points of intersection of \( C_1 \) and \( C_2 \) in the range \( -\frac{\pi}{2} < \theta \le \frac{\pi}{2} \).
(b) Sketch both curves on the same diagram, clearly labeling the points of intersection.
(c) Calculate the exact area of the region that lies inside \( C_2 \) but outside \( C_1 \). Show all your working, including the integration steps.