Moderate -0.3 This is a standard C4 harmonic form question with routine application of R cos(θ - α) = R cos α cos θ + R sin α sin θ. Students equate coefficients to find R = 5 and α = arctan(4/3), then use the range [-R, R] to find f(θ) ∈ [2, 12], making the reciprocal's maximum 1/2. While it requires multiple steps, it follows a well-practiced algorithm with no novel problem-solving, making it slightly easier than average.
Express \(3\cos\theta + 4\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\).
Hence find the range of the function \(f(\theta)\), where
$$f(\theta) = 7 + 3\cos\theta + 4\sin\theta \quad \text{for } 0 \leqslant \theta \leqslant 2\pi.$$
Write down the greatest possible value of \(\frac{1}{7 + 3\cos\theta + 4\sin\theta}\). [6]
Express $3\cos\theta + 4\sin\theta$ in the form $R\cos(\theta - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{1}{2}\pi$.
Hence find the range of the function $f(\theta)$, where
$$f(\theta) = 7 + 3\cos\theta + 4\sin\theta \quad \text{for } 0 \leqslant \theta \leqslant 2\pi.$$
Write down the greatest possible value of $\frac{1}{7 + 3\cos\theta + 4\sin\theta}$. [6]
\hfill \mbox{\textit{OCR C4 Q1 [6]}}