2026 EJC P2 Q5

Junior College 2

10 marks

Given that $x$ is sufficiently small, find the series expansion of $\tan\left(x-\dfrac\pi3\right)$ up to and including the term in $x^2$.[4]
Hence find an approximation for the value of $\tan\left(-\dfrac{7\pi}{24}\right)$, giving your answer in the form $a+b\pi+c\pi^2$, where $a$, $b$ and $c$ are exact real constants to be determined.[3]
Find the series expansion of $\ln\left|\sec\left(x-\dfrac\pi3\right)\right|$ up to and including the term in $x^2$.[3]

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Answer:(a) $-\sqrt3+4x-4\sqrt3x^2+O(x^3)$. (b) $-\sqrt3+\pi/6-\sqrt3\pi^2/144$. (c) $\ln2-\sqrt3x+2x^2+O(x^3)$.

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