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Sine & Cosine Rules (Cambridge O Level Maths)
Revision Note
Sine Rule
What is the sine rule?
- The sine rule allows us to find missing side lengths or angles in non-right-angled triangles
- It states that for any triangle with angles A, B and C
-
- Where
- is the side opposite angle A
- is the side opposite angle B
- is the side opposite angle C
- Where
- Sin 90° = 1 so if one of the angles is 90° this becomes SOH from SOHCAHTOA
How can we use the sine rule to find missing side lengths or angles?
- The sine rule can be used when you have any opposite pairs of sides and angles
- Always start by labelling your triangle with the angles and sides
- Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
- Use the formula to find the length of a side
- To find a missing angle you can rearrange the formula and use the form
- Substitute the values you have into the formula and solve
What is the ambiguous case of the sine rule?
- If the sine rule is used in a triangle given two sides and an angle which is not the angle between them there may be more than one possible triangle which could be drawn
- The side opposite the given angle could be in two possible positions
- This will create two possible values for each of the missing angles and two possible lengths for the missing side
- The two angles found opposite the given side (not the ambiguous side) will add up to 180°
- The question will usually tell you whether the angle you are looking for is acute or obtuse
- The sine rule will always give you the acute option but you can subtract from 180° to find the obtuse angle
- Sometimes the obtuse angle will not be valid as it could cause the sum of the three interior angles of the triangle to exceed 180°
Examiner Tip
- Remember to check that your calculator is in degrees mode!
- The formula for the sine rule can be found in the list of formulas on page 2
Worked example
The following diagram shows triangle ABC. , , angle .
Use the sine rule to calculate the value of:
i)
,
ii)
.
Cosine Rule
What is the cosine rule?
- The cosine rule allows us to find missing side lengths or angles in non-right-angled triangles
- It states that for any triangle
;
-
- Where
- a is the side opposite angle A
- b and c are the other two sides
- Where
- Both of these formulae are rearrangements of the same thing
- The first version is used to find a missing side
- The second version is a rearrangement of this and can be used to find a missing angle
- Cos 90° = 0 so if A = 90° this becomes Pythagoras’ Theorem
How can we use the cosine rule to find missing side lengths or angles?
- The cosine rule can be used when you have two sides and the angle between them or all three sides
- Always start by labelling your triangle with the angles and sides
- Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
- Use the formula to find an unknown side
- Use the formula to find an unknown angle
- A is the angle between sides b and c
- Substitute the values you have into the formula and solve
Examiner Tip
- Remember to check that your calculator is in degrees mode!
- The formula for the cosine rule can be found in the list of formulas on page 2
Worked example
The following diagram shows triangle . , , .
Calculate the value of angle .
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