Law of Sines vs Law of Cosines: Which One to Use
Once a triangle loses its right angle, SOHCAHTOA stops working — those ratios were built on the presence of a 90° corner. Two more general tools take over for any triangle at all: the Law of Sines and the Law of Cosines. The real skill is not memorising them but knowing which one the problem is asking for, and that comes down to reading the information you are given.
The two laws stated
Throughout, a lowercase letter is the side directly across from the uppercase angle of the same letter, so side a faces angle A, and so on.
The Law of Sines links each side to the sine of its opposite angle:
a / sin A = b / sin B = c / sin C
The Law of Cosines generalises the Pythagorean theorem to any angle:
c2 = a2 + b2 − 2ab × cos C
When C is exactly 90°, cos C is 0 and the last term vanishes, leaving plain old a2 + b2 = c2. The Law of Cosines is really Pythagoras with a correction term for the angle.
Which law fits your triangle
The deciding question is whether you already have a matched pair — an angle and the side opposite it.
- Reach for the Law of Sines when you can form such a pair: the cases ASA, AAS, and SSA.
- Reach for the Law of Cosines when you cannot: the cases SAS (two sides and the angle between them) and SSS (all three sides).
A quick way to remember it: if you are handed all three sides, or two sides hugging an angle, no ratio pairs up cleanly, so cosines it is. Anything looser usually pairs an angle with its side, so sines does the job.
The ambiguous case, briefly
The SSA arrangement — two sides and an angle not tucked between them — is the tricky one. Depending on the numbers it can describe two different triangles, exactly one, or none at all. Whenever you apply the Law of Sines to SSA, check whether a second, obtuse angle also fits, since the sine of an angle equals the sine of its supplement.
Worked example 1: Law of Sines
In a triangle, angle A = 40°, angle B = 60°, and the side opposite A is a = 10. Find side b. You have angle A paired with side a, so the Law of Sines applies:
b = a × sin B / sin A = 10 × sin 60° / sin 40° = 10 × 0.8660 / 0.6428 ≈ 13.47.
Worked example 2: Law of Cosines
Now take a triangle with sides a = 5 and b = 7 and the included angle C = 60° between them. This is SAS, so use the Law of Cosines to find side c:
c2 = 52 + 72 − 2 × 5 × 7 × cos 60° = 25 + 49 − 70 × 0.5 = 74 − 35 = 39.
So c = √39 ≈ 6.24.
Finishing what you started
Notice that a single law rarely finishes a triangle on its own. After the Law of Cosines hands you side c above, the fastest next step is usually the Law of Sines to chase down a remaining angle, since you now have the paired side and angle it needs. The two laws work as a relay team: cosines to break into an awkward triangle, sines to mop up the rest.
The Law of Cosines is also the tool for the pure SSS case. Rearranged as cos C = (a2 + b2 − c2) / (2ab), it turns three known sides directly into an angle, with no ambiguity to worry about. That freedom from the two-triangle trap is one more reason to prefer cosines whenever a matched pair is missing.
With the right law chosen, the arithmetic is short every time. Try both on your own triangles with the calculators on SinCosTan.