Find maximum or minimum value

2. $$f ( x ) = \cos x + 2 \sin x$$

1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), 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. $$g ( x ) = 3 - 7 f ( 2 x )$$
2. Using the answer to part (a),
   1. write down the exact maximum value of \(\mathrm { g } ( x )\),
   2. find the smallest positive value of \(x\) for which this maximum value occurs, giving your answer to 2 decimal places.

4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$

1. Write \(\mathrm { f } ( x )\) in the form $$a \sin 2 x + b \cos 2 x + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
2. Use the answer to part (a) to write \(\mathrm { f } ( x )\) in the form $$R \sin ( 2 x + \alpha ) + c$$ where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 significant figures.
3. Hence, or otherwise,
   1. state the maximum value of \(\mathrm { f } ( x )\)
   2. find the second smallest positive value of \(x\) at which a maximum value of \(\mathrm { f } ( x )\) occurs. Give your answer to 3 significant figures.

1 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.

1 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).

4 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.

1. (a) Express \(2 \cos \theta + 8 \sin \theta\) 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 first three terms of an arithmetic sequence are $$\cos x \quad \cos x + \sin x \quad \cos x + 2 \sin x \quad x \neq n \pi$$ Given that \(S _ { 9 }\) represents the sum of the first 9 terms of this sequence as \(x\) varies,
(b) (i) find the exact maximum value of \(S _ { 9 }\) (ii) deduce the smallest positive value of \(x\) at which this maximum value of \(S _ { 9 }\) occurs.