IB Maths AA SLTopic 3 β Geometry & TrigPaper 1 & 2~10 min read
Sine Rule, Cosine Rule & Area of a Triangle
When a triangle isn’t right-angled, SOH CAH TOA stops working. You need three new tools instead β the sine rule, the cosine rule, and the trig area formula. The hard part isn’t the formulas (they’re all in the booklet); it’s picking the right one.
π What you need to know
Naming convention: capital letters (A, B, C) for angles, lowercase (a, b, c) for the side opposite that angle.
Sine rule:asin A = bsin B = csin C β use when you have an angle paired with its opposite side.
Cosine rule:a2 = b2 + c2 β 2bc cos A β use when you have two sides and the angle between them, or all three sides.
Area of a triangle: A = 12ab sin C β use when you have two sides and the angle between them.
All three formulas are in the IB formula booklet.
The naming convention (read this first)
Every triangle problem in this note uses the same labelling: capital opposite lowercase. Side a sits opposite angle A; side b sits opposite angle B; side c sits opposite angle C. Get this wrong and every formula falls apart.
Standard triangle labelling
Quick rule: if the question gives you a side, look across to the angle directly opposite β that’s the matching pair (a β A, etc.). This pairing is what the sine rule and cosine rule are built around.
When to use each formula β the decision tree
Look at what the question gives you. The pattern of givens tells you which formula to reach for.
π Sine rule (find side)
Given: 2 angles + any 1 side (AAS or ASA).
asin A = bsin B
π Sine rule (find angle)
Given: 2 sides + 1 angle opposite one of them.
sin Aa = sin Bb
π Cosine rule (find side)
Given: 2 sides + the angle between them (SAS).
a2 = b2+c2β2bc cos A
π Cosine rule (find angle)
Given: all 3 sides (SSS).
cos A = b2+c2βa22bc
π¦ Area formula
Given: 2 sides + the angle between them.
A = 12ab sin C
π Right-angled triangle?
If the triangle has a right angle, use Pythagoras / SOH CAH TOA β they’re faster.
a2+b2=c2
The fastest way to choose: count the pieces of information you’ve got. Two angles + a side β sine rule. Two sides + the included angle β cosine rule (or area). Three sides β cosine rule (rearranged). Side and its opposite angle, plus another side or angle β sine rule.
The sine rule
The sine rule connects each side to the sine of its opposite angle. It says all three of these ratios are equal:
Sine Ruleasin A = bsin B = csin Cβ in formula booklet
You only ever use two of the three ratios at a time β the one with the unknown, and one with full information.
Flip it for angles: when finding an angle, flip the formula upside down: sin Aa = sin Bb. Same rule, easier to rearrange when the unknown is in the numerator.
β The ambiguous case
If you’re given two sides and an angle that’s not between them (an “SSA” setup), the sine rule can sometimes give two valid answers β one acute, one obtuse. If sin ΞΈ = 0.6, then ΞΈ could be 36.87Β° or 180Β° β 36.87Β° = 143.13Β°. Always check whether the obtuse answer fits the triangle (the angles must add to 180Β°).
The cosine rule
When the sine rule doesn’t apply (because you don’t have an angle paired with its opposite side), you reach for the cosine rule. There are two forms β one for finding a side, one for finding an angle:
Cosine Rule (find a side)a2 = b2 + c2 β 2bc cos Aβ in formula booklet
Cosine Rule (find an angle)
cos A = b2 + c2 β a22bcβ in formula booklet
π€ The cosine rule is just Pythagoras with a correction
When A = 90Β°, cos A = 0, so the formula becomes a2 = b2 + c2 β that’s Pythagoras! The “β2bc cos A” term is the correction for non-right-angled triangles. The cosine rule is the generalisation; Pythagoras is the special case.
π§ Memory hook for the cosine rule
The angle A in the formula is always opposite the side you’re finding or matching. So if you want side a, use angle A. If you’re given all three sides and want an angle, the formula matches up: solving for cos A uses a2 as the “subtracted” term in the numerator.
Area of a triangle
Most students know one area formula: half base times height. But that requires you to know the perpendicular height β which you usually don’t. The trig version is much more useful in IB:
Area of a Triangle (trig version)
Area = 12ab sin Cβ in formula booklet
The angle must be the one between the two sides. If you use sides a and b, then angle C (the angle between them) is the right one. Don’t grab a different angle by mistake.
You can also write this as Area = Β½ac sin B or Β½bc sin A β same formula, different pairing. Always: two sides Γ sin(included angle), all halved.
Quick reference β what to use when
Find
Given
Use
A side
2 angles + 1 side
sine rule
A side
2 sides + included angle
cosine rule
An angle
2 sides + 1 opposite angle
sine rule (flipped)
An angle
3 sides (SSS)
cosine rule (rearranged)
Area
2 sides + included angle
Β½ab sin C
Anything
right-angled triangle
SOH CAH TOA / Pythagoras
Worked examples
WE 1
Sine rule β find a side
In triangle ABC, angle A = 50Β°, angle B = 70Β°, and side a = 8 cm. Find side b.
Step 1: Pick the right pairs
Have a β A (full pair), want b with B.
Step 2: Set up sine rulebsin 70Β° = 8sin 50Β°Step 3: Solve for bb = 8 Γ sin 70Β°sin 50Β° = 8 Γ 0.9397β¦0.7660β¦b β 9.81 cm (3 s.f.)cross-multiply if rearranging confuses you!
WE 2
Sine rule β find an angle
In triangle ABC, side a = 12, side b = 15, and angle A = 40Β°. Find angle B.
Step 1: Use the flipped formsin B15 = sin 40Β°12Step 2: Solve for sin Bsin B = 15 Γ sin 40Β°12 = 15 Γ 0.6428β¦12 = 0.8035β¦Step 3: Apply inverse sineB = sinβ»ΒΉ(0.8035β¦)B β 53.5Β° (3 s.f.)check: 180 β 53.5 = 126.5Β° also works for sin, but 40+126.5 > 180 so it’s invalid here
WE 3
Cosine rule β find a side (SAS)
In triangle ABC, side b = 9, side c = 11, and angle A = 65Β°. Find side a.
Step 1: Apply the cosine rulea2 = b2 + c2 β 2bc cos AStep 2: Substitutea2 = 92 + 112 β 2(9)(11) cos 65Β°= 81 + 121 β 198 Γ 0.4226β¦= 202 β 83.68β¦ = 118.32β¦Step 3: Square roota = β118.32β¦a β 10.9 cm (3 s.f.)don’t forget the square root at the end!
WE 4
Cosine rule β find an angle (SSS)
A triangle has sides a = 7, b = 8, c = 13. Find angle C.
Step 1: Use the rearranged cosine rulecos C = a2+b2βc22abStep 2: Substitutecos C = 49 + 64 β 1692(7)(8) = β56112 = β0.5Step 3: Apply inverse cosineC = cosβ»ΒΉ(β0.5)C = 120Β°negative cosine β obtuse angle. cosβ»ΒΉ handles this automatically.
WE 5
Area using the trig formula
A triangle has two sides of length 6 cm and 9 cm with an included angle of 75Β°. Find its area to 3 s.f.
Step 1: Apply the area formulaArea = 12ab sin CStep 2: Substitute a = 6, b = 9, C = 75Β°Area = 12(6)(9) sin 75Β°= 27 Γ 0.9659β¦Area β 26.1 cm2 (3 s.f.)always check the angle is the one BETWEEN the two sides!
π‘ Top tips
Always sketch the triangle first and label every known side and angle. The picture tells you which formula to use.
Match capital and lowercase: angle A sits opposite side a. Get the labels right before you start.
Decide which formula in 5 seconds: got an angle paired with its opposite side? β sine rule. Got two sides and the angle between? β cosine rule. Got all three sides? β cosine rule rearranged.
For the cosine rule, the angle in the formula is always opposite the side you’re finding. Match them up.
Watch for the ambiguous case with sine rule when finding angles β sometimes both acute and obtuse answers fit, sometimes only one.
Calculator mode! SL trig questions for these formulas are usually in degrees β check before substituting.
For area, the angle must be the one BETWEEN the two sides β don’t pick the angle opposite one of them.
β Common mistakes
Wrong pairing. Using sin A with side b in the sine rule. The numerator and denominator must match (capital opposite lowercase).
Forgetting the square root after computing a2 in the cosine rule. The result is a2, not a.
Using the cosine rule when the sine rule is faster. If you have an angle paired with its opposite side, sine rule wins β it’s a one-step solve.
Picking the wrong angle for the area formula. The angle has to be between the two sides β not opposite either of them.
Missing the ambiguous case. When using the sine rule for an angle, both acute and obtuse answers may be valid β check the triangle inequality (angles sum to 180Β°).
Calculator in radians by mistake. sin 60Β° in radian mode gives a totally different number β always check DEG/RAD.
Rearranging errors. Don’t try to rearrange complicated cosine-rule expressions in your head β write each step.
These three formulas β sine rule, cosine rule, and area β handle every non-right-angled triangle problem you’ll meet. Combined with Pythagoras and SOH CAH TOA, you’ve now got the full toolkit for any triangle in the IB syllabus.
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