Using the sine and cosine rules | Maths | Quizalize & Oak National Academy

Starter quiz

Which of the following is the cosine rule?

$a^2+b^2=c^2$

$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$

$a^2 = b^2+c^2-2bc\cos(A)$ ✓

$\frac{1}{2}ab\sin(C)$

$\cos(A) = \frac{b^2+c^2 - a^2}{2bc}$ ✓
Which of the following is the sine rule?

$0.5\times ab \times \text{sin}C$

$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$ ✓

$a^2 = b^2+c^2-2bc\cos(A)$

$\cos(A) = \frac{b^2+c^2 - a^2}{2bc}$

$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$ ✓
Given a triangle with lengths $a=8$ cm and angle $B=36$ ° and a length $c=8$ cm, work out the missing length to 1 decimal place.

'4.9 cm' ✓
Given a triangle with lengths $a=8$ cm and angle $B=36$ ° and a length $c=10$ cm, work out the missing length to 1 decimal place.

'5.9 cm' ✓
Given a triangle with lengths $a=4$ cm, $b=6$ cm and $c=8$ cm, work out the angle $B$ to 1 decimal place.

'46.6°' ✓
Given a triangle with lengths $a=4$ cm, $b=6$ cm and $c=4$ cm, work out the angle $B$ to 1 decimal place.

'97.2°' ✓

Exit quiz

What formula is used to find the volume of a cylinder?

Volume = $\pi\times r^2\times h$ where $h$ is the height ✓

Volume = $l\times w\times h$

Volume = $\frac{h(a+b)}{2}\times l$ where $l$ is the length

Volume = $l^3$
Using Pythagoras' theorem, find the missing length when $c$ is the hypotenuse and $b$ and $a$ are the adjacent lengths.

4 cm

⇔

$a=3$ cm , $b =$ ? cm and $c=5$ cm ✓

13 cm

⇔

$a=5$ cm , $b = 12$ cm and $c=$ ? cm ✓

7 cm

⇔

$a=$ ? cm , $b = 24$ cm and $c=25$ cm ✓

15 cm

⇔

$a=8$ cm , $b =$ ? cm and $c=17$ cm ✓

60 cm

⇔

$a=11$ cm , $b =$ ? cm and $c=61$ cm ✓
Using the diagram, work out the correct height and radius.

$h$ = 35.1 cm and $r$ = 19.1 cm ✓

$h$ = 70.2 cm and $r$ = 19.1 cm

$h$ = 17.6 cm and $r$ = 9.6 cm

$h$ = 32.1 cm and $r$ = 18.1 cm
Using the diagram, work out the correct height and radius.

$h$ = 7.6 cm and $r$ = 2.5 cm

$h$ = 30.2 cm and $r$ = 10.0 cm

$h$ = 15.2 cm and $r$ = 4.9 cm ✓

$h$ = 15.2 cm and $r$ = 3.9 cm
A plane starts at City X and flies 6 km due north to City A, then turns at a bearing of 059° and travels 11.7km to City B. If the plane flew directly from City X to City B, what is the distance?

9.7 km

11.8 km

15.7 km ✓

30.4 km
A plane starts at an airport and then travels North 12 km, then North East 8.5 km. What is the bearing and distance the plane needs to travel to get back to the airport?

Bearing of 194° for 19.0 km

Bearing of 198° for 19.0 km ✓

Bearing of 204° for 19.0 km

Bearing of 224° for 22.9 km

Bearing of 304° for 22.9 km

Worksheet

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Lesson Details

Key learning points

The cosine rule is useful when you know three sides and an angle are involved
The sine rule is useful when you know two pairs of sides and angles are involved

Common misconception

Pupils may be uncertain over how to start a problem.

Encourage pupils to draw a diagram if one is not provided and then to annotate the diagram with any given information as a first step. If a diagram is given then pupils should annotate that.

Keywords

Sine rule - The sine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.
Cosine rule - The cosine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.