Maths › Trigonometry › Triangles and the sine and cosine rules
Triangles and the sine and cosine rules
Right-angled trigonometry stops at right angles; these two rules do not. Between them, the sine and cosine rules solve every triangle that can be solved, the area formula prices any triangle from two sides and a squeeze of angle, and one honest ambiguity keeps everyone on their toes.
IN THIS TOPIC
- Use the sine rule to find sides and angles, pairing each side with its opposite angle.
- Use the cosine rule from two sides and the included angle, or from all three sides.
- Handle the ambiguous case, and find areas from two sides and the included angle.
WHAT YOU PROBABLY THINK
If sin B = 0.83, then B = sin−1 0.83 and that settles it.
The sine rule
In any triangle, label each side with the lower-case letter of the angle facing it, a opposite A and so on. The sine rule says the ratio of a side to the sine of its opposite angle is the same all the way round,
and it is the tool of choice whenever the known quantities come in an opposite pair, one side with its facing angle, plus one more piece of information.
WORKED EXAMPLE
Solving a triangle from two angles and a side
In triangle ABC, A = 40°, B = 75° and a = 8 cm. Find the remaining angle and sides.
Angles first. C = 180° − 40° − 75° = 65°.
The known opposite pair is a and A, so b = 8 sin 75°/sin 40° = 12.0 cm and c = 8 sin 65°/sin 40° = 11.3 cm, both to 3 significant figures.
The largest side ends up opposite the largest angle, which is a built-in sanity check worth two seconds at the end of every triangle question.
The cosine rule and the area formula
When the known pieces refuse to form an opposite pair, two sides with the angle between them, or all three sides and no angles, the cosine rule takes over,
which is Pythagoras with a correction term for the tilt of the angle. Alongside it sits the area formula,
with C the angle squeezed between the sides a and b. Both are on the must-learn list.
WORKED EXAMPLE
Two sides and the angle between them
A triangle has sides of 5 cm and 7 cm meeting at 60°. Find the third side and the triangle's area.
Cosine rule: a2 = 52 + 72 − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39, so a = √39 = 6.24 cm.
Area = ½ × 5 × 7 × sin 60° = 15.2 cm2.
Keeping a2 = 39 exact until the last line avoids rounding damage; √39 is an acceptable exact answer wherever the question does not demand decimals.
YOUR TURN
Three sides, no angles
A triangle has sides 4 cm, 6 cm and 8 cm. Find its largest angle, before opening the working.
Show the working
The largest angle faces the longest side, 8. Rearranged cosine rule: cos θ = (42 + 62 − 82)/(2 × 4 × 6) = −12/48 = −0.25.
θ = cos−1(−0.25) = 104.5°.
The negative cosine announced the obtuse angle before the inverse button was pressed. Cosine distinguishes acute from obtuse by sign, which is a superpower sine does not have.
The ambiguous case
Sine cannot tell an angle from its supplement, since sin θ = sin (180° − θ). So when the sine rule produces sin B = 0.83, the honest conclusion is two candidates, B = 56.1° or 123.9°, and the lie this lesson opened with collapses. Whether both survive depends on the triangle. Each candidate lives only if the angles found so far leave something positive for the third angle.
TRY IT UNSEEN
Both triangles, in full
In triangle ABC, A = 40°, b = 9 cm and a = 7 cm. Find the two possible values of B, and the two possible values of C.
Show the working
Sine rule: sin B = 9 sin 40°/7 = 0.826, so B = 55.7° or its supplement 124.3°.
Check each. 40° + 55.7° leaves C = 84.3°; 40° + 124.3° leaves C = 15.7°. Both are positive, so both triangles exist.
B = 55.7° or 124.3°, C = 84.3° or 15.7°. The side a = 7 is shorter than b = 9, which is what lets it swing down on either side of the perpendicular.
Had a been 9 or longer, only the acute B would have survived. A one-line existence check after the inverse sine is where this mark hides.
THE EXAM BIT
- Label sides and angles in opposite pairs before quoting anything; a mispaired sine rule is wrong from its first line.
- Choose by information shape: an opposite pair known means sine rule; two sides and the included angle, or three sides, means cosine rule.
- After an inverse sine, write down the supplement as well and test whether each candidate leaves a positive third angle.
- Keep squared lengths exact (a2 = 39, not a = 6.244997…) until the final line, then round to the requested figures.
- For area, the angle in ½ab sin C must be the one between the two sides used; if it is not, find it first.
CHECK YOURSELF
A triangle has sides 5 cm and 7 cm meeting at an angle of 60°. Without a calculator, find the exact length of the third side and the exact area.
Show a hint
cos 60° = ½ and sin 60° = √3/2 are exact values.
Show the answer
Cosine rule: a2 = 25 + 49 − 70 × ½ = 39, so the third side is exactly √39 cm.
Area = ½ × 5 × 7 × √3/2 = 35√3/4 cm2.
Both answers are exact because 60° carries exact trig values, a theme the next lesson makes systematic.
Sine rule for opposite pairs, cosine rule for included angles or three sides.
Inverse sine always offers two angles; geometry decides how many survive.
CHECK YOUR PROGRESS
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- Use the sine rule to find sides and angles, pairing each side with its opposite angle.
- Use the cosine rule from two sides and the included angle, or from all three sides.
- Handle the ambiguous case, and find areas from two sides and the included angle.
No animated video for this topic yet; these notes stand alone.