New
New
Using the sine and cosine rules
I can use the formula for the cosine rule and the sine rule.
Lesson 9 of 11
New
New
- Year 11
Using the sine and cosine rules
I can use the formula for the cosine rule and the sine rule.
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
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.
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.
Task B Q1 can be scaffolded by asking pupils to calculate the bearing from the airport rather than to the airport. You may wish for pupils to calculate from the airport first and then calculate the bearing to the airport.
Teacher tip
Licence
This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).
Lesson video
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Prior knowledge starter quiz
6 Questions
Q1.Which of the following is the cosine rule?
Which of the following is the cosine rule?
$$a^2+b^2=c^2$$
$$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
Correct answer: $$a^2 = b^2+c^2-2bc\cos(A)$$
$$a^2 = b^2+c^2-2bc\cos(A)$$
$$\frac{1}{2}ab\sin(C)$$
Correct answer: $$ \cos(A) = \frac{b^2+c^2 - a^2}{2bc}$$
$$ \cos(A) = \frac{b^2+c^2 - a^2}{2bc}$$
Q2.Which of the following is the sine rule?
Which of the following is the sine rule?
$$0.5\times ab \times \text{sin}C$$
Correct answer: $$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
$$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
$$a^2 = b^2+c^2-2bc\cos(A)$$
$$ \cos(A) = \frac{b^2+c^2 - a^2}{2bc}$$
Correct answer: $$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$$
$$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$$
Q3.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.
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.
Correct Answer: 4.9 cm, 4.9
4.9 cm, 4.9
Q4.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.
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.
Correct Answer: 5.9 cm, 5.9
5.9 cm, 5.9
Q5.Given a triangle with lengths $$a=4$$ cm, $$b=6$$ cm and $$c=8$$ cm, work out the angle $$B$$ to 1 decimal place.
Given a triangle with lengths $$a=4$$ cm, $$b=6$$ cm and $$c=8$$ cm, work out the angle $$B$$ to 1 decimal place.
Correct Answer: 46.6°, 46.6, 46.6 degrees
46.6°, 46.6, 46.6 degrees
Q6.Given a triangle with lengths $$a=4$$ cm, $$b=6$$ cm and $$c=4$$ cm, work out the angle $$B$$ to 1 decimal place.
Given a triangle with lengths $$a=4$$ cm, $$b=6$$ cm and $$c=4$$ cm, work out the angle $$B$$ to 1 decimal place.
Correct Answer: 97.2°, 97.2, 97.2 degrees
97.2°, 97.2, 97.2 degrees
Assessment exit quiz
6 Questions
Q1.What formula is used to find the volume of a cylinder?
What formula is used to find the volume of a cylinder?
Correct answer: Volume = $$\pi\times r^2\times h$$ where $$h$$ is the height
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 = $$l^3$$
Q2.Using Pythagoras' theorem, find the missing length when $$c$$ is the hypotenuse and $$b$$ and $$a$$ are the adjacent lengths.
Using Pythagoras' theorem, find the missing length when $$c$$ is the hypotenuse and $$b$$ and $$a$$ are the adjacent lengths.
Correct Answer:4 cm,$$a=3$$ cm , $$b = $$? cm and $$c=5$$ cm
4 cm - $$a=3$$ cm , $$b = $$? cm and $$c=5$$ cm
Correct Answer:13 cm,$$a=5$$ cm , $$b = 12$$ cm and $$c=$$? cm
13 cm - $$a=5$$ cm , $$b = 12$$ cm and $$c=$$? cm
Correct Answer:7 cm ,$$a=$$? cm , $$b = 24$$ cm and $$c=25$$ cm
7 cm - $$a=$$? cm , $$b = 24$$ cm and $$c=25$$ cm
Correct Answer:15 cm ,$$a=8$$ cm , $$b = $$? cm and $$c=17$$ cm
15 cm - $$a=8$$ cm , $$b = $$? cm and $$c=17$$ cm
Correct Answer:60 cm ,$$a=11$$ cm , $$b = $$ ? cm and $$c=61$$ cm
60 cm - $$a=11$$ cm , $$b = $$ ? cm and $$c=61$$ cm
Q3.Using the diagram, work out the correct height and radius.
Using the diagram, work out the correct height and radius.
Correct answer: $$h$$ = 35.1 cm and $$r$$ = 19.1 cm
$$h$$ = 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
Q4.Using the diagram, work out the correct height and radius.
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
Correct answer: $$h$$ = 15.2 cm and $$r$$ = 4.9 cm
$$h$$ = 15.2 cm and $$r$$ = 4.9 cm
$$h$$ = 15.2 cm and $$r$$ = 3.9 cm
Q5.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?
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
Correct answer: 15.7 km
15.7 km
30.4 km
Q6.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?
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
Correct answer: Bearing of 198° 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
