4.09a Polar coordinates: convert to/from cartesian

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1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find the area of the region enclosed by the curve, giving your answer in exact form.
2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.

5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$

1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
   (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
   (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

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1. 1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
   2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
   3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
2. 1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
   2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.

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1. Given that \(\mathrm { f } ( x ) = \arccos x\),
   1. sketch the graph of \(y = \mathrm { f } ( x )\),
   2. show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\),
   3. obtain the Maclaurin series for \(\mathrm { f } ( x )\) as far as the term in \(x ^ { 3 }\).
2. A curve has polar equation \(r = \theta + \sin \theta , \theta \geqslant 0\).
   1. By considering \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) show that \(r\) increases as \(\theta\) increases. Sketch the curve for \(0 \leqslant \theta \leqslant 4 \pi\).
   2. You are given that \(\sin \theta \approx \theta\) for small \(\theta\). Find in terms of \(\alpha\) the approximate area bounded by the curve and the lines \(\theta = 0\) and \(\theta = \alpha\), where \(\alpha\) is small.

1

1. 1. Differentiate the equation \(\sin y = x\) with respect to \(x\), and hence show that the derivative of \(\arcsin x\) is \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
   2. Evaluate the following integrals, giving your answers in exact form.
      (A) \(\int _ { - 1 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x\) (B) \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x\)
2. A curve has polar equation \(r = \tan \theta , 0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The points on the curve have cartesian coordinates \(( x , y )\). A sketch of the curve is given in Fig. 1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{99f0c663-bb5b-4456-854c-df177f5d8349-2_493_796_1123_605} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Show that \(x = \sin \theta\) and that \(r ^ { 2 } = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }\).
   Hence show that the cartesian equation of the curve is $$y = \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } .$$ Give the cartesian equation of the asymptote of the curve.

5 This question concerns curves with polar equation \(r = \sec \theta + a\), where \(a\) is a constant.

1. State the set of values of \(\theta\) between 0 and \(2 \pi\) for which \(r\) is undefined. For the rest of the question you should assume that \(\theta\) takes all values between 0 and \(2 \pi\) for which \(r\) is defined.
2. Use your graphical calculator to obtain a sketch of the curve in the case \(a = 0\). Confirm the shape of the curve by writing the equation in cartesian form.
3. Sketch the curve in the case \(a = 1\). Now consider the curve in the case \(a = - 1\). What do you notice?
   By considering both curves for \(0 < \theta < \pi\) and \(\pi < \theta < 2 \pi\) separately, describe the relationship between the cases \(a = 1\) and \(a = - 1\).
4. What feature does the curve exhibit for values of \(a\) greater than 1 ? Sketch a typical case.
5. Show that a cartesian equation of the curve \(r = \sec \theta + a\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).

1

1. A curve has polar equation \(r = a ( 1 - \sin \theta )\), where \(a > 0\) and \(0 \leqslant \theta < 2 \pi\).
   1. Sketch the curve.
   2. Find, in an exact form, the area of the region enclosed by the curve.
2. 1. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x\).
   2. Find, in an exact form, the value of the integral \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).

7 A curve has polar equation \(r = 1 + \cos 3 \theta\), for \(- \pi < \theta \leqslant \pi\).

1. Show that the line \(\theta = 0\) is a line of symmetry.
2. Find the equations of the tangents at the pole.
3. Find the exact value of the area of the region enclosed by the curve between \(\theta = - \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\).

2 A curve has polar equation \(r = \cos \theta \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Find

1. the equations of the tangents at the pole,
2. the maximum value of \(r\),
3. a cartesian equation of the curve, in a form not involving fractions.

8 The equation of a curve is \(x ^ { 2 } + y ^ { 2 } - x = \sqrt { x ^ { 2 } + y ^ { 2 } }\).

1. Find the polar equation of this curve in the form \(r = \mathrm { f } ( \theta )\).
2. Sketch the curve.
3. The line \(x + 2 y = 2\) divides the region enclosed by the curve into two parts. Find the ratio of the two areas.

4 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where \(0 \leqslant \theta \leqslant \pi\).

1. Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area bounded by \(C _ { 1 } , C _ { 2 }\) and the line \(\theta = 0\).

11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(r\) decreases as \(\theta\) increases.
2. The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\), $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
3. Draw a sketch of \(C\).
4. Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).

6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$\begin{array} { l l } C _ { 1 } : & r = a \\ C _ { 2 } : & r = 2 a \cos 2 \theta , \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi \end{array}$$ where \(a\) is a positive constant. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point with polar coordinates ( \(a , \beta\) ). State the value of \(\beta\). Show that the area of the region bounded by the initial line, the arc of \(C _ { 1 }\) from \(\theta = 0\) to \(\theta = \beta\), and the arc of \(C _ { 2 }\) from \(\theta = \beta\) to \(\theta = \frac { 1 } { 4 } \pi\) is $$a ^ { 2 } \left( \frac { 1 } { 6 } \pi - \frac { 1 } { 8 } \sqrt { } 3 \right)$$

4 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\). Sketch the graph of \(C\). Find the area of the region \(R\) enclosed by \(C\) and the initial line. The half-line \(\theta = \frac { 1 } { 5 } \pi\) divides \(R\) into two parts. Find the area of each part, correct to 3 decimal places.

The curve \(C\) has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = a ^ { 2 } \left( x ^ { 2 } - y ^ { 2 } \right)$$ where \(a\) is a positive constant. Show that \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta$$ Sketch \(C\) for \(- \pi < \theta \leqslant \pi\). Find the area of the sector between \(\theta = - \frac { 1 } { 4 } \pi\) and \(\theta = \frac { 1 } { 4 } \pi\). Find the polar coordinates of all points of \(C\) where the tangent is parallel to the initial line.

10 Use the identity \(2 \sin P \cos Q \equiv \sin ( P + Q ) + \sin ( P - Q )\) to show that $$2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right) \equiv \cos \left( 2 \theta - \frac { 3 } { 4 } \pi \right) + \frac { 1 } { \sqrt { } 2 }$$ A curve has polar equation \(r = 2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(0 \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\). Sketch the curve and state the polar equation of its line of symmetry, justifying your answer. Show that the area of the region enclosed by the curve is \(\frac { 3 } { 8 } ( \pi + 1 )\).

5 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\). Draw a sketch of \(C\). Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta = \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 2 } { 3 } \pi\).

4 The curve \(C\) has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 a ^ { 2 } x y\), where \(a\) is a positive constant. Show that the polar equation of \(C\) is \(r ^ { 2 } = a ^ { 2 } \sin 2 \theta\). Sketch \(C\) for \(- \pi < \theta \leqslant \pi\). Find the area enclosed by one loop of \(C\).

5 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$\begin{array} { l l } C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi \\ C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi \end{array}$$ Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram. Find the exact value of the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the half-line \(\theta = 0\).

4 A curve \(C\) has polar equation \(r ^ { 2 } = 8 \operatorname { cosec } 2 \theta\) for \(0 < \theta < \frac { 1 } { 2 } \pi\). Find a cartesian equation of \(C\). Sketch \(C\). Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\), the half-line \(\theta = \frac { 1 } { 6 } \pi\) and the half-line \(\theta = \frac { 1 } { 3 } \pi\).
[0pt] [It is given that \(\int \operatorname { cosec } x \mathrm {~d} x = \ln \left| \tan \frac { 1 } { 2 } x \right| + c\).]

A curve \(C\) has polar equation \(r = 2 a \cos \left( 2 \theta + \frac { 1 } { 2 } \pi \right)\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Show that \(r = - 2 a \sin 2 \theta\) and sketch \(C\).
2. Deduce that the cartesian equation of \(C\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = - 4 a x y .$$
3. Find the area of one loop of \(C\).
4. Show that, at the points (other than the pole) at which a tangent to \(C\) is parallel to the initial line, $$2 \tan \theta = - \tan 2 \theta .$$

3 The curve \(C\) has polar equation \(r = \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\), showing full working.
3. Find a cartesian equation of \(C\).

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = a \\ & C _ { 2 } : r = 2 a | \cos \theta | \end{aligned}$$ where \(a\) is a positive constant. The curves intersect at the points \(P _ { 1 }\) and \(P _ { 2 }\).

1. Find the polar coordinates of \(P _ { 1 }\) and \(P _ { 2 }\).
2. In a single diagram, sketch \(C _ { 1 } , C _ { 2 }\) and their line of symmetry.
3. The region \(R\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is bounded by the \(\operatorname { arcs } O P _ { 1 } , P _ { 1 } P _ { 2 }\) and \(P _ { 2 } O\), where \(O\) is the pole. Find the area of \(R\), giving your answer in exact form.

The curve \(C _ { 1 }\) has polar equation \(r ^ { 2 } = 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the exact value of \(\theta\) at \(Q\).
3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).

2 The curve \(C\) has polar equation \(r ^ { 2 } = \ln ( 1 + \theta )\), for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Sketch \(C\).
2. Using the substitution \(u = 1 + \theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.