Small angle approximationsAQA A-Level Maths: Subtopic test

1. 1
   $\theta$ is a small angle measured in radians.

   (a)

   Find the best small angle approximation for $\cos4\theta$.

   [1 mark]

   • A$1-4\theta^2$
   • B$1-8\theta^2$
   • C$1-16\theta^2$
   • D$1-2\theta^2$

   (b)

   Find the best small angle approximation for $\sin\theta\tan2\theta$.

   [1 mark]

   • A$2\theta$
   • B$\theta^2$
   • C$3\theta^2$
   • D$2\theta^2$

   (c)

   Use small angle approximations to find an approximate value of $\frac{1-\cos2\theta}{\theta\sin\theta}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An isosceles triangle $ABC$ has $AB=AC=10$ cm and $B\hat{A}C=\theta$ radians, where $\theta$ is small.

   (a)

   Find a small angle approximation for the length $BC$.

   [1 mark]

   • A$5\theta$ cm
   • B$20\theta$ cm
   • C$10\theta$ cm
   • D$20\left(1-\frac{\theta^2}{2}\right)$ cm

   (b)

   Find a small angle approximation for the perpendicular distance from $A$ to $BC$.

   [1 mark]

   • A$10-\frac{5\theta^2}{4}$ cm
   • B$10-5\theta^2$ cm
   • C$10-\frac{\theta^2}{8}$ cm
   • D$10-\frac{5\theta^2}{2}$ cm

   (c)

   Use a small angle approximation to show that the area of triangle $ABC$ is approximately $50\theta$ cm$^2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For small $\theta$ (in radians), let $f(\theta)=\frac{1-\cos4\theta}{\theta\tan2\theta}$.

   (a)

   Show that $f(\theta)\approx4$.

   [3 marks]

   (b)

   A student uses $f(\theta)\approx4$ when $\theta=0.1$. Use a calculator to find the exact value of $f(0.1)$ and the percentage error in the student's value, relative to the exact value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A tower of height $120$ m stands on level ground. From a point $P$ on the ground at a horizontal distance $d$ m from the base of the tower, the angle of elevation of the top is $\theta$ radians, where $\theta$ is small. The straight-line distance from $P$ to the top of the tower is $L$ m.

   (a)

   (i) Show that, for small $\theta$, $d\approx\frac{120}{\theta}$.
   (ii) Estimate $d$ when $\theta=0.03$.
   (iii) Calculate $d$ exactly when $\theta=0.03$, and find the percentage error in your estimate from (ii).

   [6 marks]

   (b)

   (i) Show that $L-d\approx\frac12L\theta^2$.
   (ii) Use $\theta=0.03$ and $L\approx4000$ to estimate $L-d$.
   (iii) A surveyor takes $d\approx L$. Justify whether this is acceptable here.

   [6 marks]

   Total for question 4: 12 marks

End of questions