Key points about the cosine rule
The cosine rule is a A fact, rule, or principle that is expressed in terms of mathematical symbols. The plural of formula is formulae. used to find a missing side or angle in a triangle when two sides andAn angle between two given sides., or all the lengths of all three sides, are known.
There are two versions of the cosine rule:
- Find an unknown side using πΒ² = πΒ² + πΒ² β 2ππ cosπ΄
- Find an unknown angle using cosπ΄ = \(\frac{πΒ² + πΒ² β πΒ²}{2ππ}\)
Scientific calculators need to be used for trigonometry and should be in degrees mode. Often there is a small D or DEG at the top of the calculator screen. If not, go into the calculator settings to change the angle units to degrees.
Make sure you are confident with finding unknown sides and angles in right-angled triangles to be successful with non-right-angled A branch of mathematics which explores the relationships between sides and angles in a triangle..
How to find an unknown side using the cosine rule
To find an unknown side in a triangle, two sides and An angle between two given sides. must be known.
Label the angles and sides of the triangle and use the formula πΒ² = πΒ² + πΒ² β 2ππ cosπ΄ to find the missing side.
If the The point at which two or more lines cross. The corner of a shape. The plural form is vertices. of the triangle are not called π΄, π΅ and πΆ, it is common practice to rename them to assist with theThe process of replacing a letter (or variable) with a number. into the formula. Make sure vertex π΄ is opposite the side that needs to be calculated.
Answers should use the given notation in the question.
- An SAS (two sides and the included angle) triangle is a unique triangle which can be constructed with a pencil, ruler and protractor.
Follow the worked example below
GCSE exam-style questions
- Calculate the length of side π¦.
Give the answer to one decimal place.
π¦ = 16Β·6 cm
- Label the sides of the triangle.
Here the vertices are not labelled, so pick the vertex with angle 135Β° to be π΄. The choice of π΅ and πΆ does not matter.
The 8 cm side, opposite angle πΆ, is called π.
The 10 cm side, opposite angle π΅, is called π.
The side labelled π¦, opposite angle π΄, is called π.
- Substitute the values of π΄, π, π and π into the formula to give
π¦Β² = 10Β² + 8Β² β (2 Γ 10 Γ 8)cos(135).
- 10Β² = 100, 8Β² = 64 and 2 Γ 10 Γ 8 = 160, so this simplifies to
π¦Β² = 100 + 64 β 160cos(135)
- Type 100 + 64 - 160cos(135) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the cosine button.
Remember to close the bracket after typing in the angle.
This gives π¦Β² = 277Β·1370β¦
It is important not to round the numbers at this stage.
- The inverse of squaring is square rooting, so to find π¦, calculate the square root of 277Β·1370β¦
Type the square root button followed by the 'ANS' button into a scientific calculator.
This gives the answer of π¦ = 16Β·6474β¦
Therefore, rounded to one decimal place, π¦ = 16Β·6 cm.
- Calculate the length of ππ.
Give the answer to one decimal place.
ππ = 8Β·6 m
- Label the sides of the triangle.
Since the vertices are not called π΄, π΅ and πΆ, let vertex π, with angle 67Β°, be π΄. The choice of π΅ and πΆ doesn't matter.
Let vertex π be π΅ and vertex π be πΆ.
The 9 m side, opposite angle π΅, is called π.
The 6 m side, opposite angle πΆ, is called π.
The 10 cm side, opposite angle π΅, is called π.
The side labelled ππ, opposite angle π΄ is called π.
- Substitute the values of π΄, π, π and π into the formula to give
ππΒ² = 9Β² + 6Β² β (2 Γ 9 Γ 6)cos(67).
- 9Β² = 81, 6Β² = 36 and 2 Γ 9 Γ 6 = 108, so this simplifies to
ππΒ² = 81 + 36 β 108cos(67)
- Type 81 + 36 - 108cos(67) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the cosine button.
Remember to close the bracket after typing in the angle.
This gives ππΒ² = 74Β·8010β¦
It is important not to round the numbers at this stage.
- Find ππ by calculating the square root of 74Β·8010β¦
Type the square root button followed by the 'ANS' button into a scientific calculator.
This gives the answer of ππ = 8Β·6487β¦
Therefore, rounded to one decimal place, ππ = 8Β·6 m.
How to re-arrange the cosine formula
To use the cosine formula to find a missing angle in a triangle, the formula must be re-arranged to become:
cosπ΄ = \(\frac{πΒ² + πΒ² β πΒ²}{2ππ} \)
Find out more about re-arranging the cosine formula below
How to find an unknown angle using the cosine rule
To find an unknown angle in a triangle, the length of all three sides must be known.
Find the missing angle by labelling the angles and sides of the triangle and using the formula:
cosπ΄ = \(\frac{πΒ² + πΒ² β πΒ²}{2ππ} \)
This formula has vertex π΄ as the angle to be calculated. If the variables used for the vertices are not π΄, π΅ and πΆ, rename them to fit.
- When finding angles using trigonometry, the inverse function is used.
Follow the worked example below
GCSE exam-style questions
- Calculate the size of angle π.
Give the answer to one decimal place.
π = 102Β·6
- Label the sides of the triangle.
Since the vertices are not called π΄, π΅ and πΆ, let vertex, π, the angle to be calculated, be π΄. The choice of π΅ and πΆ doesn't matter.
Let vertex π be π΅ and vertex π be πΆ.
The 11 m side, opposite angle π΄, is called π.
The 8 m side, opposite angle π΅, is called π.
The 6 m side, opposite angle πΆ, is called π.
Substitute the values of π΄, π, π and π into the rearranged formula to give cosπ = \(\frac{8Β² + 6Β² β 11Β²}{2 Γ 8 Γ 6} \).
Work out the value of each of the squares.
8Β² = 64
6Β² = 36
11Β² = 121
- Simplify the numerator and denominator.
64 + 36 β 121 = β 212 Γ 8 Γ 6 = 96
So, cosπ = \(\frac{β 21}{96} \).
- Work out the angle, π, by using the inverse function of cosine.
π = cosβ»ΒΉ(\(\frac{β 21}{96} \))
Press 'shift' then 'cos' to write cosβ»ΒΉ on a scientific calculator.
- Type β 21 Γ· 96.
Remember to close the brackets.
This gives π = 102Β·6356β¦
Rounded to 1 decimal place, Angle π = 102Β·6Β°.
- Calculate the size of angle πΆ.
Give the answer to one decimal place.
πΆ = 33Β·4Β°
- Label the sides of the triangle.
Although the vertices are called π΄, π΅ and πΆ, the angle to be calculated is not π΄. Swap the vertices so angle πΆ is called π΄ and vice-versa.
In this case, the 4Β·2 cm side, opposite angle π΄, is called π.
The 7Β·5 cm side, opposite angle π΅, is called π.
The 7 cm side, opposite angle πΆ, is called π.
- Substitute the values of π΄, π, π and π into the rearranged formula to give
cosπ΄ = \(\frac{7Β·5Β² + 7Β² β 4Β·2Β²}{2 Γ 7Β·5 Γ 7} \).
- Work out the value of each of the squares.
7Β·5Β² = 56Β·25
7Β² = 49
4Β·2Β² = 17Β·64
- Simplify the numerator and the denominator.
56Β·25 + 49 β 17Β·64 = 87Β·61 and 2 Γ 7Β·5 Γ 7 = 105
So cosπ΄ = \(\frac{87Β·61}{105} \).
- Work out the angle, π΄, using the inverse function of cosine:
π΄ = cosβ»ΒΉ(\(\frac{87Β·61}{105}\))
Press 'Shift' then 'cos' to write cosβ»ΒΉ on a scientific calculator.
- Type 87Β·61 Γ· 105.
Remember to close the brackets.
This gives π΄ = 33Β·4486β¦
Rounded to 1 decimal place, and remembering to swap back to the original variable, Angle πΆ = 33Β·4Β°.
Check your understanding
Quiz β Cosine rule
Practise what you've learned about the cosine rule with this quiz.
Now you've revised the cosine rule, why not look at combined transformations and invariant points?
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