Small angle approximation

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

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

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

Use small angle approximations to find the small negative root of the equation $$\sin x + \cos x = 0.5.$$ [3]

Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]

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.

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.

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)

11 Show that, for the angle \(45 ^ { \circ }\), the formula \(\sin \theta \approx \frac { 4 \theta ( 180 - \theta ) } { 40500 - \theta ( 180 - \theta ) }\) given in line 28 gives the same approximation for the sine of the angle as the formula \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) given in line 23.

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 .

12

1. Show that \(\cos x = \sin \left( x + \frac { \pi } { 2 } \right)\).
2. Hence show that \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) gives the approximation \(\cos x \approx \frac { \pi ^ { 2 } - 4 x ^ { 2 } } { \pi ^ { 2 } + x ^ { 2 } }\), as stated in line 31. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.
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1. Given that \(\theta\) is small, show that \(2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2\). [2]
2. Hence, when \(\theta\) is small, show that $$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$ where \(a\), \(b\) are constants to be found. [4]

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

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]

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]