The forms R cos(theta ± a) and R sin(theta ± a)Edexcel International A Level Maths: Subtopic test

1. 1
   The expression $3\sin\theta+4\cos\theta$ is written in the form $R\sin(\theta+\alpha)$, where $R>0$ and $0<\alpha<90^{\circ}$.

   (a)

   Find the value of $R$.

   [1 mark]

   • A$7$
   • B$5$
   • C$25$
   • D$\sqrt7$

   (b)

   Find the value of $\alpha$.

   [1 mark]

   • A$36.9^{\circ}$
   • B$45.0^{\circ}$
   • C$53.1^{\circ}$
   • D$143.1^{\circ}$

   (c)

   Hence state the maximum value of $3\sin\theta+4\cos\theta$ and the smallest positive value of $\theta$ at which it occurs, to 1 decimal place.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\mathrm{f}(\theta)=\cos\theta-\sqrt3\sin\theta$, with $\theta$ in radians. It is written in the form $R\cos(\theta+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$.

   (a)

   Find the value of $R$.

   [1 mark]

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

   (b)

   Find the value of $\alpha$.

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{\pi}{6}$
   • C$\frac{\pi}{4}$
   • D$\frac{2\pi}{3}$

   (c)

   Hence find the minimum value of $\mathrm{f}(\theta)$ and the smallest positive value of $\theta$ at which it occurs.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are measured in degrees and a calculator may be used.

   (a)

   Express $5\cos\theta-12\sin\theta$ in the form $R\cos(\theta+\alpha)$, where $R>0$ and $0<\alpha<90^{\circ}$. Give $\alpha$ to 1 decimal place.

   [3 marks]

   (b)

   Hence solve $5\cos\theta-12\sin\theta=6$ for $0\le\theta<360^{\circ}$, giving your answers to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water, $D$ metres, at a harbour entrance $t$ hours after midnight is modelled by $D=6+2\cos\left(\frac{\pi t}{6}\right)+3\sin\left(\frac{\pi t}{6}\right)$, for $0\le t\le24$. Angles are in radians and a calculator may be used.

   (a)

   (i) Express $2\cos x+3\sin x$ in the form $R\sin(x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$. Give $\alpha$ to 3 significant figures.
   (ii) Hence find the maximum depth of water and the time, to 2 decimal places, at which it first occurs after midnight.

   [6 marks]

   (b)

   A ship can enter the harbour only when $D\ge9$. Find the times between 00:00 and 06:00 at which the depth is at least $9$ m, giving your answers to the nearest minute.

   [6 marks]

   Total for question 4: 12 marks

End of questions