The Sine Rule, Cosine Rule and Area of a Triangle
The sine and cosine rules extend trigonometry from right-angled triangles to any triangle. The sine rule, a/sin A = b/sin B = c/sin C, is used when a side and its opposite angle are known together with one other side or angle. The cosine rule, a squared = b squared + c squared - 2bc cos A, is used when three sides are known, or when two sides and the angle between them are known. The area of a triangle can be found without its perpendicular height using one half ab sin C, where C is the angle between the two named sides. Labelling matters: a lower-case letter is the side opposite the angle with the matching capital letter.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Label the triangle first: capital letters at the vertices, and each lower-case letter on the side directly opposite the matching capital.
- Choose the rule by what you have. A matching side and opposite angle points to the sine rule. Three sides, or two sides with the angle between them, points to the cosine rule.
- Write the rule in the form that puts the unknown on top before substituting, so the rearrangement happens once and cleanly.
- For an unknown angle with the cosine rule, use the rearranged form cos A = (b squared + c squared - a squared) divided by 2bc, then take the inverse cosine.
- Keep full accuracy in the calculator through every intermediate step and round only at the end, since rounding an angle early can shift a final length noticeably.
- For area, use one half ab sin C only when C is the angle BETWEEN sides a and b. If the known angle is elsewhere, find the required angle first.
Worked example
In triangle ABC, b = 7 cm, c = 9 cm and angle A = 52 degrees. Find the length of side a, and then the area of the triangle. Give both answers to 3 significant figures.
- Two sides and the angle between them are known, so use the cosine rule: a squared = b squared + c squared - 2bc cos A.
- Substitute: a squared = 7 squared + 9 squared - 2(7)(9) cos 52 degrees = 49 + 81 - 126 cos 52 degrees.
- Evaluate: cos 52 degrees = 0.61566, so 126 times 0.61566 = 77.57, giving a squared = 130 - 77.57 = 52.43.
- Take the square root: a = 7.240..., so a = 7.24 cm to 3 significant figures.
- For the area use one half bc sin A, since A is the angle between sides b and c: area = 0.5 times 7 times 9 times sin 52 degrees.
- Evaluate: 0.5 times 63 = 31.5, and sin 52 degrees = 0.78801, so the area = 31.5 times 0.78801 = 24.8 cm squared to 3 significant figures.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Which rule would you use given two angles and one side?Show answer
Answer: The sine rule, because a side and its opposite angle can be paired.
Q2Write the cosine rule rearranged to find angle A.Show answer
Answer: cos A = (b squared + c squared - a squared) divided by 2bc.
Q3In triangle ABC, which side is opposite angle B?Show answer
Answer: Side b.
Q4Find the area of a triangle with sides 8 cm and 5 cm and an included angle of 30 degrees.Show answer
Answer: 0.5 times 8 times 5 times sin 30 = 20 times 0.5 = 10 cm squared.
Q5Why should you avoid rounding intermediate values?Show answer
Answer: Rounding early carries an error into every later step, which can change the final answer at the required accuracy.
Q6A triangle has sides 5, 6 and 7. Which rule finds its largest angle?Show answer
Answer: The cosine rule, since all three sides are known. The largest angle is opposite the longest side, 7.
Q7Can one half ab sin C be used if C is not between sides a and b?Show answer
Answer: No. The formula requires the angle to be the included angle between the two sides used, so a different angle must be found first.
Exam-style questions
Written in the style of a IGCSE Maths exam paper, with a full mark scheme.
In triangle PQR, PQ = 11 cm, QR = 8 cm and PR = 14 cm. Find the size of angle Q, giving your answer to 1 decimal place.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
A triangular field ABC has angle A = 43 degrees, angle B = 68 degrees and side AB = 120 m. Find the area of the field to the nearest square metre.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 6 available
Free printable worksheet
Want more practice on paper? Download the the sine rule, cosine rule and area of a triangle worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
Next topics
Not quite what you needed?
Tell us what is missing on the sine rule, cosine rule and area of a triangle, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.