Estimate root of equation

A question is this type if and only if it asks to estimate the value of a root α of an equation where α is stated to be small, using small angle approximations.

3 questions · Standard +0.1

1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x
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Edexcel Paper 1 2019 June Q2
5 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-04_670_1447_212_333} \captionsetup{labelformat=empty} \caption{Figure 1}
\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}
Edexcel Paper 1 2023 June Q4
5 marks Standard +0.3
  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.
  • WJEC Unit 3 Specimen Q1
    4 marks Standard +0.3
    Find a small positive value of \(x\) which is an approximate solution of the equation. $$\cos x - 4\sin x = x^2.$$ [4]