Cosine rule
Apply the cosine rule to find unknown sides in non-right-angled triangles; rearrange to find unknown angles using .
Worked examples
Finding an unknown side
Straightforward
Problem
In triangle , cm, cm and . Find side .
1
Write down the cosine rule for side .
2
Substitute the known values: , , .
3
Evaluate each term, using .
4
Take the square root.
Answer
Side is approximately 8.29 cm.
Finding an unknown angle
Moderate
Problem
In triangle , m, m and m. Find angle .
1
The angle is opposite side . Rearrange the cosine rule to find .
2
Substitute the known values.
3
The negative value tells us is obtuse. Find the angle.
Answer
Angle is approximately 92.1°.
Real-world application
Challenging
Problem
From a lighthouse , boat is 24 km away on a bearing of 035° and boat is 18 km away on a bearing of 105°. Find the distance between the two boats.
1
Find the angle at the lighthouse. The two bearings differ by .
2
Apply the cosine rule to triangle with km, km and included angle .
3
Evaluate, using .
4
Take the square root to find .
Answer
The distance between the two boats is approximately 24.59 km.
Practise
Q1·Straightforward
In triangle , cm, cm and . Use the cosine rule to find side . Give your answer to 2 decimal places.
Explanation
Applying the cosine rule:
Q2·Straightforward
In triangle , m, m and . Find side using the cosine rule. Give your answer to 2 decimal places.
Explanation
Applying the cosine rule:
Q3·Straightforward
In triangle , cm, cm and . Find side using the cosine rule. Give your answer to 2 decimal places.
Explanation
Applying the cosine rule:
Q4·Straightforward
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Explanation
Rearranging the cosine rule to find angle :
Q5·Moderate
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Explanation
Finding angle :
The negative cosine tells us is obtuse, which makes sense as the side cm is the longest.
The negative cosine tells us is obtuse, which makes sense as the side cm is the longest.
Q6·Moderate
In triangle , cm, cm and . Find side to 2 decimal places.
Explanation
Applying the cosine rule:
Q7·Moderate
Two ships leave port at the same time. Ship sails 15 km on a bearing of 040° and ship sails 20 km on a bearing of 100°. The angle between their paths is 60°. Find the distance between the two ships to 2 decimal places.
Explanation
The two paths form a triangle with the port at the included angle of 60°.
Applying the cosine rule:
Applying the cosine rule:
Q8·Moderate
A triangular garden has sides 10 m, 12 m and 16 m. Find the largest angle of the garden to 1 decimal place.
Explanation
The largest angle is opposite the longest side, so let , , .
The garden has an obtuse angle of about 92.9°.
The garden has an obtuse angle of about 92.9°.
Q9·Moderate
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Explanation
Finding angle :
Q10·Challenging
Two roads leave an intersection. Road A runs 12 km to town and road B runs 18 km to town . The angle between the roads at the intersection is 85°. Find the straight-line distance from town to town to 2 decimal places.
Explanation
Let the intersection be . We have km, km, and .
Applying the cosine rule:
Applying the cosine rule:
Q11·Challenging
From town , town is 25 km away on a bearing of 048° and town is 38 km away on a bearing of 130°. Find the distance to 2 decimal places.
Explanation
The angle between the two bearings at is .
Applying the cosine rule:
Applying the cosine rule:
Q12·Challenging
A triangular block of land has sides 32 m, 45 m and 60 m. Find the angle opposite the 60 m side to 1 decimal place.
Explanation
Let m (the longest side), m, m.
Open Math
Cosine rule
Measurement · MS-M6
Name:
Date:
Q1Straightforward
In triangle , cm, cm and . Use the cosine rule to find side . Give your answer to 2 decimal places.
Q2Straightforward
In triangle , m, m and . Find side using the cosine rule. Give your answer to 2 decimal places.
Q3Straightforward
In triangle , cm, cm and . Find side using the cosine rule. Give your answer to 2 decimal places.
Q4Straightforward
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Q5Moderate
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Q6Moderate
In triangle , cm, cm and . Find side to 2 decimal places.
Q7Moderate
Two ships leave port at the same time. Ship sails 15 km on a bearing of 040° and ship sails 20 km on a bearing of 100°. The angle between their paths is 60°. Find the distance between the two ships to 2 decimal places.
Q8Moderate
A triangular garden has sides 10 m, 12 m and 16 m. Find the largest angle of the garden to 1 decimal place.
Q9Moderate
In triangle , cm, cm and cm. Find angle to 1 decimal place.
Q10Challenging
Two roads leave an intersection. Road A runs 12 km to town and road B runs 18 km to town . The angle between the roads at the intersection is 85°. Find the straight-line distance from town to town to 2 decimal places.
Q11Challenging
From town , town is 25 km away on a bearing of 048° and town is 38 km away on a bearing of 130°. Find the distance to 2 decimal places.
Q12Challenging
A triangular block of land has sides 32 m, 45 m and 60 m. Find the angle opposite the 60 m side to 1 decimal place.
Worked solutions and answers at openmath.au/year-12/standard-2/non-right-angled-trigonometry/cosine-rule