2022 ASRJC P1 Q4
(i)
Show that the first two non-zero terms of the Maclaurin series for $\tan \theta $ is given by $\theta +\frac{1}{3}{{\theta }^{3}}$. You may use the standard results given in the List of Formulae (MF26).
[2]
In the right-angle triangle $OBC$ shown above, point $A$ lies on $OB$ such that $OA=1$, $OB=x$, where $x>1$ and $OC=1$. It is given that angle $COB$ is $\frac{\pi }{2}$ radians and that angle $ACB$ is $\theta $ radians (see diagram).
(ii)
Show that $AB=\frac{2\tan \theta }{1-\tan \theta }$.
[2]
(iii)
Given that $\theta $ is a sufficiently small angle, show that
$AB\approx a\theta +b{{\theta }^{2}}+c{{\theta }^{3}}$
for exact real constants $a$, $b$ and $c$ to be determined.
[3]
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