Watch out
Pairing a side with the wrong angle.
Back to National 5 Mathematics
SQA National 5 Mathematics
Sine rule
Solving non-right-angled triangles with matching angle-side pairs.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Check the triangle is not right-angled.
- Look for a matching side-angle pair.
- Decide whether you are finding a side or an angle.
Explanation
The sine rule is used in non-right-angled triangles when you have a matching side and opposite angle pair.
Use a / sin A = b / sin B when finding a side, or sin A / a = sin B / b when finding an angle.
Angles and their opposite sides must match across the triangle.
Visual support
Sine rule pairs
Key formulae and rules
- a / sin A = b / sin B = c / sin C
- sin A / a = sin B / b
Check
Check your answer against the diagram. The longest side should still be opposite the largest angle.
Exam tip
Circle a known matching pair before writing the sine rule.
Calculator tip
Make sure the calculator is in degree mode before using sin, cos, tan or inverse trig.
Worked examples
Worked example 1
Find a side
In triangle ABC, A = 40 degrees, B = 65 degrees and side a = 8 cm. Find side b
- Use b / sin 65 = 8 / sin 40.
- b = 8 sin 65 / sin 40
- b = 11.28..
Answer: b = 11.3 cm to 1 decimal place
Worked example 2
Find an angle
In triangle ABC, a = 7 cm, b = 10 cm and A = 35 degrees. Find angle B
- sin B / 10 = sin 35 / 7
- sin B = 10 sin 35 / 7
- B = sin−1(0.819...)
So: B = 55.0 degrees to 1 decimal place
Worked example 3
Find a third angle first
A = 50 degrees, C = 80 degrees, side a = 12 cm. Find side c.
- A and side a form a matching pair.
- c / sin 80 = 12 / sin 50
- c = 12 sin 80 / sin 50
Answer: c = 15.4 cm to 1 decimal place
Watch out
- Pairing a side with the wrong angle.
- Using sine rule when there is no matching pair.
- Rounding an angle before using it again.