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.

5 questions

Edexcel F2 2024 January Q4
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. 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.
  2. 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.
Edexcel F2 2016 June Q4
4. $$f ( x ) = \sin \left( \frac { 3 } { 2 } x \right)$$
  1. 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 \(\left( x - \frac { \pi } { 3 } \right) ^ { 4 }\)
  2. Hence obtain an estimate of \(\sin \frac { 1 } { 2 }\), giving your answer to 4 decimal places.
Edexcel F2 2023 June Q6
  1. 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.
  2. 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.
  3. Use the answer to part (b) to determine, to four significant figures, an approximate value of \(\sec \left( \frac { 7 \pi } { 24 } \right)\)
Edexcel F2 2021 October Q5
5. Given that \(y = \tan ^ { 2 } x\)
  1. 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.
  2. 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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Edexcel F2 2018 Specimen Q5
  1. Given that \(y = \cot x\),
    1. show that
    $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \cot x + 2 \cot ^ { 3 } x$$
  2. 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.
  3. 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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