A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 5: Radians · Lesson 4

Small angle approximations

111 marks

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Quick questions to start.

A1
Needed today
Expand $(1 - \frac{θ ^{2}}{2})^{2}$ , ignoring any terms in $θ^{3}$ or higher powers.
[2 marks]
Answer
A2
Needed today
Write $3^{\circ}$ in radians, as a multiple of $π$ .
[2 marks]
Answer
A3
Needed today
Find the first three terms, in ascending powers of $θ$ , of the binomial expansion of $(1 + 2 θ)^{- 1}$ .
[2 marks]
Answer
A4
Common slip
Work out $sin 0.1$ to 5 decimal places. Make sure your calculator is in radian mode.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use $sin θ \approx θ$ to estimate $sin 0.2$ . Find the percentage error in this estimate, to 3 significant figures.

Hint The percentage error is (estimate − true value) ÷ true value × 100. Use the true value from your calculator, in radian mode.

B1
Use $sin θ \approx θ$ to estimate $sin 0.15$ . Find the percentage error in this estimate, to 3 significant figures.
[4 marks]
Answer

Example 2

Given that $θ$ is small, show that $\frac{1 - cos θ}{θ sin θ} \approx \frac{1}{2}$ .

Hint Replace

cos θ

1 - \frac{θ ^{2}}{2}

and

sin θ

θ

, then simplify.

B2
Given that $θ$ is small, show that $\frac{1 - cos 2 θ}{θ tan θ} \approx 2$ .
[4 marks]

Example 3

Given that $θ$ is small, find an approximation for $3 cos 2 θ + sin θ$ in the form $a + b θ + c θ^{2}$ .

Hint Replace

2 θ

as a whole:

cos 2 θ \approx 1 - \frac{( 2 θ ) ^{2}}{2} = 1 - 2 θ^{2}

B3
Given that $θ$ is small, find an approximation for $4 cos 3 θ - 2 tan θ$ in the form $a + b θ + c θ^{2}$ .
[4 marks]
Answer

Example 4

Given that $θ$ is small, use the small angle approximations to find an approximate solution of $2 sin θ + cos θ = 1.1$ , to 3 significant figures.

Hint The approximations turn the equation into a quadratic in

θ

. It has two roots: keep the one that is small.

B4
Given that $θ$ is small, use the small angle approximations to find an approximate solution of $3 tan θ + 2 cos θ = 2.1$ , to 3 significant figures.
[5 marks]
Answer

Example 5

In triangle $O A B$ , $O A = 5$ cm, $O B = 8$ cm and angle $A O B = θ$ radians, where $θ$ is small. Show that $A B^{2} \approx 9 + 40 θ^{2}$ .

Hint Use the cosine rule, then replace

cos θ

1 - \frac{θ ^{2}}{2}

B5
In triangle $O P Q$ , $O P = 3$ cm, $O Q = 7$ cm and angle $P O Q = θ$ radians, where $θ$ is small. Find an approximation for $P Q^{2}$ in the form $a + b θ^{2}$ .
[5 marks]
Answer

Practice

A-Level

C1
Amy says that $sin 4^{\circ} \approx 4$ , because $sin θ \approx θ$ for small angles. Explain her mistake. Then find the percentage error in using $sin θ \approx θ$ for an angle of $4^{\circ}$ , to 3 significant figures.
[4 marks]
Answer

C2
Ben writes: " $cos 3 θ \approx 1 - \frac{3 θ ^{2}}{2}$ when $θ$ is small." Explain his mistake, give the correct approximation, and use it to estimate $cos 0.3$ .
[4 marks]
Answer
C3
Use $cos θ \approx 1 - \frac{θ ^{2}}{2}$ to estimate $cos 1$ . Find the percentage error in this estimate, to 3 significant figures, and explain why the error is large.
[5 marks]
Answer
C4
Given that $θ$ is small, show that $\frac{sin θ tan θ}{1 - cos θ} \approx 2$ .
[4 marks]
C5
Use a small angle approximation to estimate $cos 5^{\circ}$ , to 6 decimal places. Compare it with the value on your calculator.
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
This question is about the expression $cos 4 θ + 2 sin 3 θ$ , where $θ$ is small and in radians.
(a)
Show that $cos 4 θ + 2 sin 3 θ \approx 1 + 6 θ - 8 θ^{2}$ .
[3 marks]
(b)
Use the approximation with $θ = 0.1$ to estimate $cos 0.4 + 2 sin 0.3$ .
[2 marks]
Answer
(c)
Find the percentage error in your estimate, to 3 significant figures.
[2 marks]
Answer
(d)
Use the approximation to find, to 3 significant figures, the small value of $θ$ for which $cos 4 θ + 2 sin 3 θ = 1.2$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Kim writes: "When $θ$ is small, $cos 2 θ \approx 1 - 2 θ$ , so $\frac{1 - cos 2 θ}{sin θ} \approx \frac{2 θ}{θ} = 2$ ."
Kim is wrong.
(a)
Explain the mistake Kim has made.
[2 marks]
(b)
Find a correct approximation for $\frac{1 - cos 2 θ}{sin θ}$ in terms of $θ$ .
[3 marks]
Answer
(c)
Hence estimate $\frac{1 - cos 0.1}{sin 0.05}$ .
[2 marks]
Answer
(d)
Find the percentage error in your estimate in part (c), to 3 significant figures.
[2 marks]
Answer
Total for D2: 9 marks
D3
Given that $θ$ is small, find the value that $\frac{cos 2 θ - cos 3 θ}{θ tan θ}$ is close to. Show your working.
[6 marks]
Answer
D4
In triangle $O A B$ , $O A = 10$ cm, $O B = 12$ cm and angle $A O B = θ$ radians, where $θ$ is small.
(a)
Show that $A B^{2} \approx 4 + 120 θ^{2}$ .
[3 marks]
(b)
Use the approximation to estimate $A B$ when $θ = 0.1$ , to 4 significant figures.
[2 marks]
Answer
(c)
Find the percentage error in your estimate of $A B$ , to 2 significant figures.
[2 marks]
Answer
(d)
Use the approximation to find the value of $θ$ for which $A B = 3$ cm, to 3 significant figures.
[2 marks]
Answer
Total for D4: 9 marks
D5
A tower $T F$ is $30$ m tall and stands on level ground. From a point $P$ on the ground, the angle of elevation of the top of the tower $T$ is $0. 6^{\circ}$ . The distance from $P$ to the foot of the tower is $d$ m.
(a)
Write $0. 6^{\circ}$ in radians, to 3 significant figures.
[2 marks]
Answer
(b)
Use $tan θ \approx θ$ to estimate $d$ , to the nearest metre.
[2 marks]
Answer
(c)
Show that the percentage error in this estimate is less than $0.01%$ .
[3 marks]
(d)
Explain why the same method would give a much worse estimate if the angle of elevation were $2 5^{\circ}$ .
[2 marks]
Total for D5: 9 marks
D6
This question is about the expression $\frac{2 - cos θ}{1 + sin θ}$ , where $θ$ is small and in radians.
(a)
Show that $\frac{2 - cos θ}{1 + sin θ} \approx 1 - θ + \frac{3}{2} θ^{2}$ .
[4 marks]
(b)
Use the approximation to estimate the value of the expression when $θ = 0.1$ .
[2 marks]
Answer
(c)
Find the percentage error in your estimate, to 3 significant figures.
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A footbridge is an arc of a circle of radius $400$ m. Its two ends, $A$ and $B$ , are level and $40$ m apart. The height $h$ m is the height of the middle of the bridge above the straight line $A B$ . Use a small angle approximation to show that $h \approx \frac{c ^{2}}{8 r}$ , where $c$ is the length $A B$ and $r$ is the radius, and hence estimate $h$ .
[9 marks]
Hint 1 · What to try
Let the angle $A O B$ at the centre be $2 α$ . Then $h = r - r cos α$ , since the middle of $A B$ is $r cos α$ from the centre.
Hint 2 · The first line
$α$ is small, so $cos α \approx 1 - \frac{α ^{2}}{2}$ , which gives $h \approx \frac{r α ^{2}}{2}$ .
Hint 3 · The full method
Half the chord is $r sin α$ , so $\frac{c}{2} \approx r α$ , which gives $α \approx \frac{c}{2 r}$ . Put this into $h$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2 when θ is small and measured in radians.
I can replace kθ by the whole of kθ in an approximation, such as cos 3θ ≈ 1 − 9θ²/2.
I can find an approximate expression and its percentage error.
I can use the approximations to solve an equation or to simplify the cosine rule.

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