Show that the first two non-zero terms of the Maclaurin series for $\tan \theta $ is given by

Question

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). Show that $AB=\frac{2\tan \theta }{1-\tan \theta }$.[2] 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]

Accepted Answer

(i) $\tan\theta\approx\theta+\frac13\theta^3$. (ii) $AB=\frac{2\tan\theta}{1-\tan\theta}$. (iii) $AB\approx2\theta+2\theta^2+\frac83\theta^3$, so $a=2$, $b=2$, and $c=\frac83$.

2022 ASRJC P1 Q4

2022 ASRJC P1 Q4

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