9.
$$\mathrm { f } ( \theta ) = 5 \cos \theta - 4 \sin \theta \quad \theta \in \mathbb { R }$$
- 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,
- describe fully the transformation that is a stretch,
- describe fully the transformation that is a translation.
Given
$$g ( \theta ) = \frac { 90 } { 4 + ( f ( \theta ) ) ^ { 2 } } \quad \theta \in \mathbb { R }$$
- find the range of g.
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