1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x

1

1. For a small angle \(\theta\), where \(\theta\) is in radians, show that \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } \approx 3 - 2 \theta\).
2. Hence determine an approximate solution to \(2 \cos \theta + ( 1 - \tan \theta ) ^ { 2 } = 28 \sin \theta\).

1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of

$$\frac { \theta \tan 3 \theta } { \cos 2 \theta - 1 }$$

1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of

$$\frac { 1 - \cos 4 \theta } { 2 \theta \sin 3 \theta }$$ (3)

2. \begin{figure}[h] \end{figure} Figure 1 shows a plot of part of the curve with equation \(y = \cos x\) where \(x\) is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.

1. Use Diagram 1 to show why the equation $$\cos x - 2 x - \frac { 1 } { 2 } = 0$$ has only one real root, giving a reason for your answer. Given that the root of the equation is \(\alpha\), and that \(\alpha\) is small,
2. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.
   \includegraphics[max width=\textwidth, alt={}]{91a2f26a-add2-4b58-997d-2ae229548217-05_664_1452_246_333}
   \section*{Diagram 1}

1. The curve \(C\), in the standard Cartesian plane, is defined by the equation

$$x = 4 \sin 2 y \quad \frac { - \pi } { 4 } < y < \frac { \pi } { 4 }$$ The curve \(C\) passes through the origin \(O\)

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the origin.
2. 1. Use the small angle approximation for \(\sin 2 y\) to find an equation linking \(x\) and \(y\) for points close to the origin.
   2. Explain the relationship between the answers to (a) and (b)(i).
3. Show that, for all points \(( x , y )\) lying on \(C\), $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { a \sqrt { b - x ^ { 2 } } }$$ where \(a\) and \(b\) are constants to be found.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 1 } { 2 } \cos x\)
• the curve has a stationary point with \(x\) coordinate \(\alpha\)
• \(\alpha\) is small
  1. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.

The point \(P ( 0,3 )\) lies on \(C\)

Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

1. Given that \(\theta\) is small and in radians, use the small angle approximations to find an approximate numerical value of

$$\frac { \theta \tan 2 \theta } { 1 - \cos 3 \theta }$$

1. Given that \(\theta\) is small and measured in radians, use the small angle approximations to show that

$$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.

1. (a) Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that

$$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a).
Adele's working is shown below. Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\) (b) (i) Identify the mistake made by Adele in her working.
(ii) Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) (2)

8 The equation of a curve is \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), where \(x\) is in radians.

1. Show that, for small values of \(x , y \approx 2 \sqrt { x } + 2 - 4 x ^ { 2 }\). The diagram shows the region bounded by the curve \(\mathrm { y } = \sqrt { \sin 4 \mathrm { x } } + 2 \cos 2 \mathrm { x }\), the axes and the line \(x = 0.1\). \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-07_499_881_589_223}
2. In this question you must show detailed reasoning. Use the approximation in part (a) to estimate the area of this region.

3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h] \end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).

9 Show that \(\mathrm { y } = \mathrm { x }\) has the same gradient as \(\mathrm { y } = \sin \mathrm { x }\) when \(\mathrm { x } = 0\), as stated in line 5 .

10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$

1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]

   \(n\)	0	1	2	3	4	5	6	7
   \(\mathrm { x } _ { \mathrm { n } }\)	1	0.958509	0.950084	0.948261	0.94786	0.947772	0.947753	0.947748

   \captionsetup{labelformat=empty} \caption{Fig. 10.1}
   \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]

   \(x\)	\(y\)
   0.9477475	\(- 7.79967 \mathrm { E } - 07\)
   0.9477485	\(- 2.90821 \mathrm { E } - 06\)
   \(x\)	\(y\)
   0.947745	\(4.54066 \mathrm { E } - 06\)
   0.947755	\(- 1.67417 \mathrm { E } - 05\)

   \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{table}
3. Consider the information in Fig. 10.1 and Fig. 10.2.

3 Use small angle approximations to estimate the solution of the equation \(\frac { \cos \frac { 1 } { 2 } \theta } { 1 + \sin \theta } = 0.825\), if \(\theta\) is small enough to neglect terms in \(\theta ^ { 3 }\) or above.

6 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-06_463_702_264_685} The diagram shows the curve \(C\) with parametric equations $$x = \frac { 1 } { 4 } \sin t , \quad y = t \cos t$$ where \(0 \leqslant t \leqslant k\).

1. Find the value of \(k\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) in terms of \(t\). The maximum point on \(C\) is denoted by \(P\).
3. Using your answer to part (ii) and the standard small angle approximations, find an approximation for the \(x\)-coordinate of \(P\).
4. (a) Show that the area of the finite region bounded by \(C\) and the \(x\)-axis is given by $$b \int _ { 0 } ^ { a } t ( 1 + \cos 2 t ) \mathrm { d } t$$ where \(a\) and \(b\) are constants to be determined.
   (b) In this question you must show detailed reasoning. Hence find the exact area of the finite region bounded by \(C\) and the \(x\)-axis.

4 For a small angle \(\theta\), where \(\theta\) is in radians, show that \(1 + \cos \theta - 3 \cos ^ { 2 } \theta \approx - 1 + \frac { 5 } { 2 } \theta ^ { 2 }\).

15

1. Show that $$\sin x - \sin x \cos 2 x \approx 2 x ^ { 3 }$$ for small values of \(x\).
   15
2. Hence, show that the area between the graph with equation $$y = \sqrt { 8 ( \sin x - \sin x \cos 2 x ) }$$ the positive \(x\)-axis and the line \(x = 0.25\) can be approximated by $$\text { Area } \approx 2 ^ { m } \times 5 ^ { n }$$ where \(m\) and \(n\) are integers to be found.
   15
3. (i) Explain why $$\int _ { 6.3 } ^ { 6.4 } 2 x ^ { 3 } \mathrm {~d} x$$ is not a suitable approximation for $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ Question 15 continues on the next page 15 (c) (ii) Explain how $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ may be approximated by $$\int _ { a } ^ { b } 2 x ^ { 3 } \mathrm {~d} x$$ for suitable values of \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-31_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{042e248a-9efa-4844-957d-f05715900ffc-36_2486_1719_221_150}

6

1. Find the first two terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 1 - \frac { x } { 2 } \right) ^ { \frac { 1 } { 2 } }$$ 6
2. Hence, for small values of \(x\), show that $$\sin 4 x + \sqrt { \cos x } \approx A + B x + C x ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found.

4 Given that \(\theta\) is a small angle, find an approximation for \(\cos 2 \theta\) Circle your answer. \(1 - \frac { \theta ^ { 2 } } { 2 }\) \(2 - 2 \theta ^ { 2 }\) \(1 - 2 \theta ^ { 2 }\) \(1 - \theta ^ { 2 }\)

When \(\theta\) is small, find an approximation for \(\cos 3\theta + \theta \sin 2\theta\), giving your answer in the form \(a + b\theta^2\) [3 marks]

Using small angle approximations, show that for small, non-zero, values of \(x\) $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where \(A\) is a constant to be determined. [4 marks]

Show that, for small values of \(x\), the graph of \(y = 5 + 4\sin\frac{x}{2} + 12\tan\frac{x}{3}\) can be approximated by a straight line. [3 marks]

\includegraphics{figure_9} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l\). P is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.

1. Show that $$\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2 = \frac{2g}{l}\cos\theta + k_1,$$ where \(k_1\) is a constant to be determined in terms of \(g\), \(l\) and \(\alpha\). [4]
2. Show that $$T = 3mg\cos\theta + k_2,$$ where \(k_2\) is a constant to be determined in terms of \(m\), \(g\) and \(\alpha\). [3]

It is given that \(\alpha\) is small enough for \(\alpha^2\) to be negligible.

1. Find, in terms of \(m\) and \(g\), the approximate tension in the string. [2]
2. Show that the motion of P is approximately simple harmonic. [3]