Standard +0.8 This question combines harmonic form (standard A-level technique) with finding the range of a rational function involving the squared harmonic expression. Part (a) is routine, but part (b) requires recognizing that f(θ)² has range [0, R²], then carefully inverting the rational function to find g's range—a multi-step problem requiring solid algebraic manipulation and understanding of function composition beyond typical textbook exercises.
9.
$$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$
a. 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.
Given that
$$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$
b.find the range of \(g\).
Score for either end achieved by a correct method. Look for \(\frac{135}{3}\) (implied by 33.75), \(\frac{135}{4 + \text{their }(\sqrt{41})^2}\) e.g. 33.75 or g...3
A1
Accept equivalent ways of writing the interval such as \([3, 33.75]\). Condone \(3 \leq g(x) \leq 33.75\) or \(3 \leq y \leq 33.75\)
(5 marks)
**Part a.** | (3 marks)
**B1** | $R = \sqrt{41}$ | Do not allow decimals for this mark. Example: $\sqrt{4^2 + 5^2} = \sqrt{41}$
**M1** | $\tan \alpha = \pm\frac{5}{4}$, $\tan \alpha = \pm\frac{4}{5}$ $\Rightarrow \alpha = \cdots$
If R is used to find $\alpha$ accept $\sin \alpha = \pm\frac{5}{R}$ or $\cos \alpha = \pm\frac{4}{R}$ $\Rightarrow \alpha = \cdots$
Example: $4 \cos \theta + 5 \sin \theta \equiv R \cos(\theta - \alpha)$ $\Rightarrow R \cos \theta \cos \alpha - \sin \theta \sin \alpha$ $\Rightarrow R \cos \alpha = 4$ $\Rightarrow \cos \alpha = \pm\frac{4}{R}$, $R \sin \alpha = 5$ $\Rightarrow \sin \alpha = \pm\frac{5}{R}$
**A1** | $\alpha = $ awrt 0.896
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**Part b.** | (2 marks)
**M1** | Score for either end achieved by a correct method. Look for $\frac{135}{3}$ (implied by 33.75), $\frac{135}{4 + \text{their }(\sqrt{41})^2}$ e.g. 33.75 or g...3
**A1** | Accept equivalent ways of writing the interval such as $[3, 33.75]$. Condone $3 \leq g(x) \leq 33.75$ or $3 \leq y \leq 33.75$
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9.
$$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$
a. 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.
Given that
$$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$
b.find the range of $g$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q9 [5]}}