Taylor series about π/3 or π/6

Questions asking for Taylor series expansion about x = π/3 or x = π/6, typically involving sin, cos, tan, sec, or cot functions.

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Determine, in ascending powers of $\left( x - \frac { \pi } { 6 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 6 } \right) ^ { 3 }$, the Taylor series expansion about $\frac { \pi } { 6 }$ of $$y = \tan \left( \frac { 3 x } { 2 } \right)$$ giving each coefficient in simplest form.
Hence show that $$\tan \frac { 3 \pi } { 8 } \approx 1 + \frac { \pi } { 4 } + \frac { \pi ^ { 2 } } { A } + \frac { \pi ^ { 3 } } { B }$$ where $A$ and $B$ are integers to be determined.

Find the Taylor series expansion for $\mathrm { f } ( x )$ about $\frac { \pi } { 3 }$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 4 }$
Hence obtain an estimate of $\sin \frac { 1 } { 2 }$, giving your answer to 4 decimal places.

Given that $y = \sec x$
1. show that
$$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sec x \tan x \left( p \sec ^ { 2 } x + q \right)$$ where $p$ and $q$ are integers to be determined.
Hence determine the Taylor series expansion about $\frac { \pi } { 3 }$ of sec $x$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$, up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 3 }$, giving each coefficient in simplest form.
Use the answer to part (b) to determine, to four significant figures, an approximate value of $\sec \left( \frac { 7 \pi } { 24 } \right)$

show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \tan x \sec ^ { 2 } x \left( p \sec ^ { 2 } x + q \right)$$ where $p$ and $q$ are integers to be determined.
Hence determine the Taylor series expansion about $\frac { \pi } { 3 }$ of $\tan ^ { 2 } x$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 3 }$, giving each coefficient in simplest form.

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Given that $y = \cot x$,
1. show that
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \cot x + 2 \cot ^ { 3 } x$$
Hence show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = p \cot ^ { 4 } x + q \cot ^ { 2 } x + r$$ where $p , q$ and $r$ are integers to be found.
Find the Taylor series expansion of $\cot x$ in ascending powers of $\left( x - \frac { \pi } { 3 } \right)$ up to and including the term in $\left( x - \frac { \pi } { 3 } \right) ^ { 3 }$.

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