Law of Sines and Cosines: When to Use Each
The law of sines and the law of cosines solve any triangle - but which do you use? Here's the simple rule, plus a worked example of each.
The law of sines (a/sin A = b/sin B = c/sin C) and the law of cosines (c² = a² + b² − 2ab·cos C) let you solve any triangle, not just right-angled ones. The rule for choosing is simple: use the law of sines when you know an angle and the side opposite it; use the law of cosines when you know two sides and the angle between them, or all three sides.
This guide states both laws, gives you a decision table for which to use, and works a full example of each - plus a warning about the one case that catches students out.
What the two laws say
The law of sines pairs each side with the sine of its opposite angle, and says those ratios are all equal:
The law of cosines links all three sides to one angle - it's the Pythagorean theorem with a correction term for non-right angles:
When to use each (the part that trips students up)
Match the law to what the problem gives you:
| When you know… | Case | Use |
|---|---|---|
| Two angles and any side | AAS / ASA | Law of sines |
| Two sides and a non-included angle | SSA (ambiguous) | Law of sines |
| Two sides and the angle between them | SAS | Law of cosines |
| All three sides | SSS | Law of cosines |
Worked example: the law of sines
Suppose angle A = 30°, angle B = 45°, and the side opposite A is a = 10. To find side b (opposite B), pair each side with its opposite angle and solve:
Worked example: the law of cosines
Now suppose you know two sides and the angle between them: a = 5, b = 7, and the included angle C = 60°. Find the third side c:
Watch out for the ambiguous case (SSA)
When you're given two sides and an angle that isn't between them (SSA), the information can describe zero, one, or two different triangles - which is why it's called the ambiguous case. If you solve for an angle with the law of sines, remember that an obtuse partner angle (180° minus the acute one) may also be valid. Always check whether the second angle still lets the triangle's angles add to less than 180°; if it does, there are two answers.
Common mistakes (and how to avoid them)
- Reaching for the law of sines when you only have SAS or SSS - you can't pair an angle with its opposite side, so use the law of cosines.
- Forgetting the ambiguous case in SSA and reporting only one triangle when two exist.
- Putting the calculator in the wrong mode - confirm degrees vs radians before you start.
- Mislabeling which angle is 'opposite' which side - the angle and side must face each other.
- Rounding sine and cosine values too early; keep a few decimal places until the final step.
How a tutor makes it click
The hard part of these problems is rarely the arithmetic - it's reading the triangle and choosing the right law. Working live on a shared whiteboard, a tutor can sketch each figure, label what's known, and coach the AAS/SSA/SAS/SSS decision until it's automatic, then keep you moving through the rest of trigonometry. Our online trigonometry tutoring pairs you with a specialist who makes the choice feel obvious.
Because the two laws unlock every triangle - not just right ones - getting them solid opens up a huge slice of trigonometry and physics. Your first trial lesson is free.
The bottom line
Two laws solve any triangle. Use the law of sines (a/sin A = b/sin B = c/sin C) when you can pair an angle with its opposite side; use the law of cosines (c² = a² + b² − 2ab·cos C) for two sides and the included angle, or all three sides. Identify the case, pick the matching law, mind the ambiguous SSA case, and the triangle falls into place.
Frequently asked questions
When do you use the law of sines vs the law of cosines?+
Use the law of sines when you know an angle and the side opposite it (plus one more angle or side) - that is, the AAS, ASA, or SSA cases. Use the law of cosines when you know two sides and the angle between them (SAS) or all three sides (SSS). A quick test: if you can pair an angle with its opposite side, reach for the law of sines; if you only have sides around an angle, use the law of cosines.
What is the law of sines?+
The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same ratio: a/sin A = b/sin B = c/sin C. You use it to find a missing side or angle when you already know an angle and the side across from it.
What is the law of cosines?+
The law of cosines relates all three sides of a triangle to one angle: c² = a² + b² − 2ab·cos C. It's a generalization of the Pythagorean theorem (when C = 90°, cos C = 0 and it reduces to c² = a² + b²). Use it for two sides and the included angle, or to find an angle when you know all three sides.
What is the ambiguous case (SSA)?+
The SSA case - two sides and an angle not between them - is called ambiguous because the given information can describe zero, one, or two different triangles. When you solve it with the law of sines, always check whether a second valid angle (180° minus the one you found) also fits, since sometimes both do.
Can the law of cosines find an angle?+
Yes. Rearranged, it solves for an angle when you know all three sides: cos C = (a² + b² − c²) / (2ab). Compute the right-hand side, then take the inverse cosine. This is the standard way to find angles in the SSS case.
Can a tutor help me with trigonometry?+
Yes - trig is highly visual, and a tutor working live on a shared whiteboard can sketch each triangle, label what's known, and coach which law a problem calls for until choosing it is automatic. That guided practice is what turns the law of sines and cosines from formulas you half-remember into tools you use with confidence.
Ready to put this into practice?
Work 1-on-1 with a vetted iTutorzz tutor - your first trial lesson is free.