Convert to Cartesian (trigonometric)

6 The parametric equations of a circle are
\(x = 2 \cos \theta - 3\) and \(y = 2 \sin \theta + 1\).
Determine the cartesian equation of the circle in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } + ( \mathrm { y } - \mathrm { b } ) ^ { 2 } = \mathrm { k }\), where \(a , b\) and \(k\) are integers.

5 A curve is defined by the parametric equations $$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$

1. 1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$ where \(a\) and \(b\) are integers.
   2. Find the gradient of the normal to the curve at the point where \(\theta = \frac { \pi } { 6 }\).
2. Show that the cartesian equation of the curve can be expressed as $$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where \(p\) is a rational number.

2. The curve \(C\) is described by the parametric equations $$x = 3 \cos t , \quad y = \cos 2 t , \quad 0 \leq t \leq \pi .$$

1. Find a cartesian equation of the curve \(C\).
2. Draw a sketch of the curve \(C\).

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).

1. Find the value of \(\theta\) at \(P\).
2. Show that the normal to the curve at \(P\) passes through the origin.
3. Find a cartesian equation for the curve.
   7. continued

7. A curve has parametric equations $$x = 3 \cos ^ { 2 } t , \quad y = \sin 2 t , \quad 0 \leq t < \pi$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 } { 3 } \cot 2 t\).
2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
3. Show that the tangent to the curve at the point where \(t = \frac { \pi } { 6 }\) has the equation $$2 x + 3 \sqrt { 3 } y = 9$$
4. Find a cartesian equation for the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
   7. continued
   7. continued

1 The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2 \theta , y = \cos \theta$$ a) Find the Cartesian equation of \(C\).
b) Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac { \pi } { 3 }\), and at the point \(Q\), where \(\theta = \frac { \pi } { 2 }\). Write down the coordinates of \(P\) and \(Q\).
c) Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. Prove by contradiction that, for every real number \(x\) such that \(0 \leqslant x \leqslant \frac { \pi } { 2 }\), $$\sin x + \cos x \geqslant 1$$ a) Given that \(f\) is a function,
i) state the condition for \(f ^ { - 1 }\) to exist,
ii) find \(f f ^ { - 1 } ( x )\).
b) The functions \(g\) and \(h\), are given by $$\begin{aligned} & g ( x ) = x ^ { 2 } - 1 \\ & h ( x ) = \mathrm { e } ^ { x } + 1 \end{aligned}$$ i) Suggest a domain for \(g\) such that \(g ^ { - 1 }\) exists.
ii) Given the domain of \(h\) is ( \(- \infty , \infty\) ), find an expression for \(h ^ { - 1 } ( x )\) and sketch, using the same axes, the graphs of \(h ( x )\) and \(h ^ { - 1 } ( x )\). Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
iii) Determine an expression for \(g h ( x )\) in its simplest form. a) Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 360 ^ { \circ }\) satisfying $$8 \sin \theta - 15 \cos \theta - 7 = 0$$ c) Determine the greatest value and the least value of the expression $$\frac { 1 } { 8 \sin \theta - 15 \cos \theta + 23 }$$ Evaluate a) \(\int _ { 1 } ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x\).
b) \(\int _ { 0 } ^ { 1 } \frac { 2 + x } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x\). The variable \(y\) satisfies the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 5 - 2 y , \quad \text { where } x \geqslant 0$$ Given that \(y = 1\) when \(x = 0\), find an expression for \(y\) in terms of \(x\). a) Differentiate the following functions with respect to \(x\), simplifying your answer wherever possible. i) \(e ^ { 3 \tan x }\),
ii) \(\frac { \sin 2 x } { x ^ { 2 } }\).
b) A function is defined implicitly by $$3 x ^ { 2 } y + y ^ { 2 } - 5 x = 5$$ Find the equation of the normal at the point (1, 2). By drawing suitable graphs, show that \(x - 1 = \cos x\) has only one root. Starting with \(x _ { 0 } = 1\), use the Newton-Raphson method to find the value of this root correct to two decimal places.

6. A curve has parametric equations $$x = 2 \cot t , \quad y = 2 \sin ^ { 2 } t , \quad 0 < t \leq \frac { \pi } { 2 } .$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of the parameter \(t\).
2. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 4 }\).
3. Find a cartesian equation of the curve in the form \(y = \mathrm { f } ( x )\). State the domain on which the curve is defined.

6 A curve is given by the parametric equations $$x = \cos \theta \quad y = \sin 2 \theta$$

1. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\).
      (2 marks)
   2. Find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\).
2. Show that the cartesian equation of the curve can be written as $$y ^ { 2 } = k x ^ { 2 } \left( 1 - x ^ { 2 } \right)$$ where \(k\) is an integer.

1 A curve is defined by the parametric equations $$x = \cos \theta \text { and } y = \sin \theta \quad \text { where } 0 \leq \theta \leq 2 \pi$$ Which of the options shown below is a Cartesian equation for this curve?
Circle your answer. $$\frac { y } { x } = \tan \theta \quad x ^ { 2 } + y ^ { 2 } = 1 \quad x ^ { 2 } - y ^ { 2 } = 1 \quad x ^ { 2 } y ^ { 2 } = 1$$