Small angle approximationsEdexcel A-Level Maths: Subtopic test

1. 1
   In this question $\theta$ is small and measured in radians, so the standard small angle approximations may be used.

   (a)

   Use a small angle approximation to estimate $\cos0.2$.

   [1 mark]

   • A$0.98$
   • B$0.96$
   • C$0.9$
   • D$0.2$

   (b)

   Which expression approximates $\sin3\theta\tan2\theta$?

   [1 mark]

   • A$5\theta^2$
   • B$6\theta^2$
   • C$6\theta$
   • D$\theta^2$

   (c)

   Calculate the percentage error in the estimate $\cos0.2\approx0.98$, using the calculator value of $\cos0.2$. Give your answer to 2 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For small $x$ in radians, consider the expression $E=\frac{1-\cos4x}{x\tan2x}$.

   (a)

   Which expression is the small angle approximation for $1-\cos4x$?

   [1 mark]

   • A$16x^2$
   • B$2x^2$
   • C$8x^2$
   • D$4x$

   (b)

   Which expression is the small angle approximation for $x\tan2x$?

   [1 mark]

   • A$2x$
   • B$x^2$
   • C$4x^2$
   • D$2x^2$

   (c)

   Hence find the approximate value of $E$ when $x$ is small.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A pendulum of length $1.5$ m swings through a small angle $\theta$ radians from the vertical. The horizontal displacement of the bob from the vertical is $d=1.5\sin\theta$ m and its height above its lowest point is $h=1.5(1-\cos\theta)$ m.

   (a)

   Use a small angle approximation to show that, for small $\theta$, $h\approx0.75\theta^2$.

   [3 marks]

   (b)

   When $\theta=0.1$, use small angle approximations to estimate $d$ and $h$. Using a calculator to find the exact value of $h$, calculate the percentage error in your estimate of $h$, to 2 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question $\theta$ and $x$ are small angles measured in radians, so the standard small angle approximations may be used.

   (a)

   (i) Show that, for small $\theta$, $4\cos\theta+3\sin\theta\approx4+3\theta-2\theta^2$.
   (ii) Hence find an approximate value of the small positive solution of $4\cos\theta+3\sin\theta=4.2$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   (i) Find an approximation, valid for small $x$, for $\frac{\cos5x-1}{x\sin2x}$.
   (ii) Evaluate $\frac{\cos5x-1}{x\sin2x}$ when $x=0.01$, giving your answer to 4 significant figures, and comment on your answer to (i).

   [6 marks]

   Total for question 4: 12 marks

End of questions