Transformations of trigonometric graphs

A question is this type if and only if it asks to describe geometric transformations (stretches, translations) that map one trigonometric curve to another using harmonic form.

6 questions · Standard +0.5

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Edexcel P3 2020 January Q9
7 marks Standard +0.3
9. $$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$
  1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. The curve with equation \(y = \cos \theta\) is transformed onto the curve with equation \(y = \mathrm { f } ( \theta )\) by a sequence of two transformations. Given that the first transformation is a stretch and the second a translation,
    1. describe fully the transformation that is a stretch,
    2. describe fully the transformation that is a translation. Given $$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
  3. find the range of g.
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    Q9

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OCR C3 2006 June Q8
11 marks Standard +0.3
8
  1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\).
  3. Solve, for \(0 ^ { \circ } < x < 360 ^ { \circ }\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1 ^ { \circ }\).
OCR C3 2014 June Q9
12 marks Standard +0.8
9
  1. Express \(5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence
    (a) give details of the transformations needed to transform the curve \(y = 5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) to the curve \(y = \sin \theta\),
    (b) find the smallest positive value of \(\beta\) satisfying the equation $$5 \cos \left( \frac { 1 } { 3 } \beta - 40 ^ { \circ } \right) + 3 \cos \left( \frac { 1 } { 3 } \beta + 20 ^ { \circ } \right) = 3 .$$ \section*{END OF QUESTION PAPER}
OCR H240/01 2019 June Q9
11 marks Standard +0.3
9
  1. Express \(3 \cos 3 x + 7 \sin 3 x\) in the form \(R \cos ( 3 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. Give full details of a sequence of three transformations needed to transform the curve \(y = \cos x\) to the curve \(y = 3 \cos 3 x + 7 \sin 3 x\).
  3. Determine the greatest value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
  4. Determine the least value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
OCR MEI Paper 1 2019 June Q10
7 marks Standard +0.3
10
  1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).
AQA Paper 2 2018 June Q8
10 marks Challenging +1.2
8
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt { 3 } \sin x - 3 \cos x + 4\) Fully justify your answer.
    8
  2. (ii) Find the greatest value of \(\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }\)