The R-formula (a cos + b sin)AQA A-Level Maths: Subtopic test

1. 1
   Let $f(\theta)=3\cos\theta+4\sin\theta$, with $\theta$ in degrees. It is written in the form $r\cos(\theta-\alpha)$, where $r>0$ and $0^\circ<\alpha<90^\circ$.

   (a)

   Find the value of $r$.

   [1 mark]

   • A$5$
   • B$7$
   • C$25$
   • D$1$

   (b)

   Find the value of $\alpha$.

   [1 mark]

   • A$36.9^\circ$
   • B$53.1^\circ$
   • C$126.9^\circ$
   • D$-53.1^\circ$

   (c)

   Write down the maximum value of $f(\theta)$ and the smallest positive value of $\theta$ at which it occurs.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $g(\theta)=\sqrt3\sin\theta-\cos\theta$, with $\theta$ in degrees. It is written in the form $R\sin(\theta-\alpha)$, where $R>0$ and $0^\circ<\alpha<90^\circ$.

   (a)

   Find the value of $R$.

   [1 mark]

   • A$4$
   • B$\sqrt2$
   • C$2$
   • D$\sqrt3-1$

   (b)

   Find the value of $\alpha$.

   [1 mark]

   • A$60^\circ$
   • B$-30^\circ$
   • C$45^\circ$
   • D$30^\circ$

   (c)

   Solve $g(\theta)=1$ for $0^\circ\le\theta\le360^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $h(x)=5\sin x+12\cos x$, where $x$ is in degrees.

   (a)

   Express $h(x)$ in the form $R\sin(x+\alpha)$, where $R>0$ and $0^\circ<\alpha<90^\circ$.

   [3 marks]

   (b)

   Hence solve $5\sin x+12\cos x=6.5$ for $0^\circ\le x<360^\circ$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water, $D$ metres, in a harbour $t$ hours after midnight is modelled by $D=10+3\cos(30t)^\circ+4\sin(30t)^\circ$ for $0\le t\le12$.

   (a)

   (i) Write $3\cos(30t)^\circ+4\sin(30t)^\circ$ in the form $R\cos(30t-\alpha)^\circ$.
   (ii) Hence find the maximum depth of water, and the first time after midnight at which it occurs, to 2 decimal places.

   [6 marks]

   (b)

   Large ships can enter the harbour only when $D\ge12$. Find the total length of time, during the 12 hours, when large ships cannot enter.

   [6 marks]

   Total for question 4: 12 marks

End of questions