10 The diagram shows a sector of a circle \(O A B\).
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-16_758_796_360_623}
The point \(C\) lies on \(O B\) such that \(A C\) is perpendicular to \(O B\).
Angle \(A O B\) is \(\theta\) radians.
10
- Given the area of the triangle \(O A C\) is half the area of the sector \(O A B\), show that
$$\theta = \sin 2 \theta$$
10
- Use a suitable change of sign to show that a solution to the equation
$$\theta = \sin 2 \theta$$
lies in the interval given by \(\theta \in \left[ \frac { \pi } { 5 } , \frac { 2 \pi } { 5 } \right]\)
10- The Newton-Raphson method is used to find an approximate solution to the equation
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| \(\theta = \sin 2 \theta\) |
10- Using \(\theta _ { 1 } = \frac { \pi } { 5 }\) as a first approximation for \(\theta\) apply the Newton-Raphson method twice
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10
(ii) Explain how a more accurate approximation for \(\theta\) can be found using the Newton-Raphson method.
10(iii) Explain why using \(\theta _ { 1 } = \frac { \pi } { 6 }\) as a first approximation in the Newton-Raphson method
[0pt]
[2 marks] does not lead to a solution for \(\theta\).