Can I use the law of sines on a right triangle?

Yes. The law of sines holds for every triangle, including right triangles, because sin 90° = 1 simply makes the hypotenuse its own ratio. That said, for a right triangle the basic SOH-CAH-TOA trig ratios or the Pythagorean theorem are usually quicker.

When should I use the law of sines vs. the law of cosines?

Use the law of sines when you have a side paired with its opposite angle (the AAS, ASA, and SSA cases). Use the law of cosines when you have two sides and the included angle (SAS) or all three sides (SSS), where no such pair exists to start from.

How do I find an unknown side using the law of sines?

Set up the ratio a/sin A = b/sin B using one known side–angle pair and the angle opposite the side you want. Solve by cross-multiplying: the missing side equals the known side times the sine of its opposite angle, divided by the sine of the known angle.

How do I find an unknown angle using the law of sines?

Use the reciprocal form sin A/a = sin B/b. Rearrange to sin B = b · sin A / a, compute the value, then take the inverse sine to get B. Remember to check whether 180° − B is also a valid second solution in the ambiguous SSA case.

What is the ambiguous case, and how do I know if there are two triangles?

The ambiguous (SSA) case is when two sides and a non-included angle can describe 0, 1, or 2 triangles. Compute the height h = b · sin A. For an acute angle A, there are two triangles when h b and none otherwise.

Why does the law of sines work?

Drop an altitude h from one vertex of the triangle. It equals both b · sin A and a · sin B, so b · sin A = a · sin B, which rearranges to a/sin A = b/sin B. Repeating with another altitude extends the equality to the third side, giving the full rule.

Law of Sines Calculator

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Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Is the Law of Sines?

The law of sines states that in any triangle, the ratio of each side to the sine of its opposite angle is the same for all three pairs. In other words, a side and the angle across from it always stay in proportion. This single relationship lets you solve a triangle — find every missing side and angle — whenever you know one side together with the angle opposite it, plus one more piece of information. It works for any oblique triangle, not just right triangles.

The rule (also called the sine rule) is written in two equivalent forms. Use the first form when you are solving for a side, and the reciprocal form when you are solving for an angle:

a/sin A = b/sin B = c/sin C

sin A/a = sin B/b = sin C/c

Here lowercase a, b, c are the side lengths and uppercase A, B, C are the angles directly opposite them — side a faces angle A, and so on. You only ever use two of the three ratios at a time: pick the pair that contains the value you know and the value you want. Because the sine function appears throughout, it helps to be comfortable with the sine, cosine, and tangent functions before you start.

When to Use the Law of Sines (AAS, ASA, SSA)

To use the law of sines you need at least one complete side–opposite-angle pair — a side and the angle facing it. That requirement is met by exactly three of the standard triangle setups. The table summarizes them; each is explained below.

Case	What you know	Solutions
AAS	Two angles and a non-included side	Exactly 1
ASA	Two angles and the included side	Exactly 1
SSA	Two sides and a non-included angle	0, 1, or 2 (ambiguous)

AAS (Angle-Angle-Side)

You know two angles and a side that is not between them. Because the two angles immediately give the third (the angles of a triangle add to 180°), you have a side and its opposite angle straight away. AAS always produces exactly one triangle.

ASA (Angle-Side-Angle)

You know two angles and the side included between them. Find the third angle first by subtracting the two known angles from 180°. That third angle is opposite your known side, giving you a complete pair to work from. ASA also always produces exactly one triangle.

SSA (Side-Side-Angle) — The Ambiguous Case

You know two sides and an angle opposite one of them, but not the angle between them. This is the famous ambiguous case: the same three measurements can describe 0, 1, or 2 different triangles. Suppose you know an acute angle A, side a opposite it, and side b. Compute the height h = b · sin A and compare:

0 triangles if a < h (side a is too short to reach the base).
1 triangle if a = h (a right triangle), or if a ≥ b.
2 triangles if h < a < b.

If angle A is 90° or more, the rule is simpler: there is exactly 1 triangle when a > b, and none otherwise, because an angle of 90° or more must sit opposite the longest side.

The calculator above handles this automatically: it solves Triangle 1 and, when a second solution exists, lists Triangle 2 with B₂ = 180° − B₁. It also flags the no-triangle conditions (sin B > 1, or A + B of 180° or more) so you never get a misleading answer. For a full walk-through of the method, see the ambiguous case of the Law of Sines explained.

How to Use This Calculator

Pick the matching tab. Choose AAS/ASA when you know two angles and a side, or SSA when you know two sides and an angle opposite one of them.
Enter your three known values. Type the two angles and side (AAS/ASA), or the two sides and angle (SSA), into the labeled fields. Angles are in degrees; sides use any consistent unit.
Press Calculate. The calculator solves the triangle instantly and fills in all six parts — three sides and three angles.
Read the results grid. Every missing side and angle appears at the top, rounded to four decimal places.
Open Step-by-Step. Expand the steps panel to see the formula with your numbers substituted in, plus a sense-check that the largest angle sits opposite the longest side.
Check the diagram. A to-scale triangle drawn from your actual inputs shows the shape. For an ambiguous SSA problem with two valid solutions, the diagram draws Triangle 1 and the results list Triangle 2.

In a hurry? Tap an example chip to load a ready-made problem and watch the calculator work before entering your own values.

How to Find a Missing Side or Angle (Worked Examples)

These static derivations show the work the same way the step-by-step panel does, so you can follow the method by hand. The calculator rounds to four decimal places, and so do we.

Find a Missing Side (AAS Example)

Given. A = 40°, B = 70°, and side a = 10 (opposite A).
Third angle. C = 180° − 40° − 70° = 70°.
Set up for b. Use a/sin A = b/sin B, so b = a · sin B / sin A = 10 · sin 70° / sin 40°.
Solve. b = 10 · 0.9397 / 0.6428 ≈ 14.6190.
Solve for c. c = a · sin C / sin A = 10 · sin 70° / sin 40° ≈ 14.6190 (equal to b, since C = B).

Find a Missing Angle (Using the Reciprocal Form)

Given. a = 8, b = 10, and A = 40° (the angle opposite a).
Set up for B. Use sin A/a = sin B/b, so sin B = b · sin A / a = 10 · sin 40° / 8.
Compute the sine. sin B = 10 · 0.6428 / 8 = 0.8035.
Take the inverse sine. B = sin⁻¹(0.8035) ≈ 53.46°.
Heads-up. Because a < b, this is an SSA problem and a second angle (126.54°) is also possible — see the ambiguous case below.

Solve the Ambiguous SSA Case (Two Triangles)

Given. a = 8, b = 10, A = 40°. Height test: h = b · sin A = 10 · 0.6428 = 6.4279. Since h < a < b, expect two triangles.
First angle. sin B = b · sin A / a = 0.8035, so B₁ = sin⁻¹(0.8035) ≈ 53.46°.
Second angle. B₂ = 180° − 53.46° = 126.54°. Since A + B₂ = 166.54° < 180°, it is valid.
Triangle 1. C₁ = 180° − 40° − 53.46° = 86.54°, then c₁ = a · sin C₁ / sin A = 8 · sin 86.54° / sin 40° ≈ 12.4230.
Triangle 2. C₂ = 180° − 40° − 126.54° = 13.46°, then c₂ = 8 · sin 13.46° / sin 40° ≈ 2.8978.
No-triangle test. Had the inputs been a = 5, b = 10, A = 40°, then sin B = 1.2856 > 1 — impossible — so no triangle exists.

Law of Sines vs. Law of Cosines

Reach for the law of sines whenever you have a side paired with its opposite angle — that is, the AAS, ASA, and SSA cases. Reach for the law of cosines when you do not have such a pair: the SAS (two sides and the included angle) and SSS (all three sides) cases. In those setups there is no ready side–angle ratio to start from, so the cosine rule is the only direct route.

Use the law of sines when…	Use the law of cosines when…
You know a side and its opposite angle (AAS, ASA, SSA)	You know SAS — two sides and the angle between them
You are finding a missing side or angle from that pair	You know SSS — all three sides and need an angle

A practical tip: switching to the law of cosines does not make the SSA ambiguity go away. For SSA the cosine rule turns into a quadratic equation for the missing side, and it has two valid answers exactly when two triangles exist. What does help is using both laws on SAS and SSS problems: find the largest angle with the law of cosines first (arccos handles obtuse angles correctly), then finish with the law of sines. The full decision guide, with worked examples, is in Law of Sines vs. Law of Cosines: when to use each.

Real-World Applications

The law of sines turns up wherever angles are easier to measure than distances:

Surveying and triangulation. Surveyors measure a baseline and the two angles to a far landmark, then use the sine rule to find distances that cannot be paced out directly.
Navigation. Ships and aircraft fix their position from the bearings of two known points, solving the resulting triangle for range and heading.
Astronomy. The parallax method finds the distance to a nearby star from the tiny angle it shifts against the background over Earth's orbit — the same angle-and-baseline idea on a cosmic scale.
Engineering and forces. Resolving forces in a truss or cable system often produces an oblique triangle of vectors that the law of sines untangles.
Architecture and design. Roof pitches, ramps, and non-rectangular layouts rely on the rule to convert known angles into precise lengths.

Mini-scenario: To find the width of a river, plant a stake on your bank and pace a 50 m baseline along it. Sight a tree on the far bank: the angle at your first stake is 38° and at the far end of the baseline is 65°. The third angle is 77°, and the law of sines turns those two angles and the 50 m baseline into the distance from your first stake to the tree: 50 · sin 65° / sin 77° ≈ 46.5 m. Multiply by sin 38° and you get the river’s width, about 28.6 m — no swimming required.

Sense-check every answer: In any triangle the largest angle sits opposite the longest side, and the smallest angle opposite the shortest side. If your solution breaks that rule, you have an error — recheck which angle pairs with which side.

Common Mistakes

Mistake — Pairing a side with the wrong angleEach ratio must use a side and the angle directly across from it. Side a goes with angle A, not with an adjacent angle. Match every side to its opposite angle before writing the proportion.

Mistake — Ignoring the second SSA solutionIn the ambiguous case, the inverse sine returns only the acute angle. Always check whether B₂ = 180° − B₁ also gives a valid triangle (when A + B₂ < 180°), or you will miss the second answer.

Mistake — Using the law of sines for SAS or SSSIf you have no side–opposite-angle pair — only two sides and the included angle, or all three sides — the sine rule cannot start. Switch to the law of cosines for those cases.

Mistake — Calculator set to radiansIf you compute angles by hand, confirm your calculator is in degree mode. sin 40° ≈ 0.6428, but sin(40 radians) is a completely different value and will wreck the result.

Mistake — Reporting an impossible answerIf sin B comes out greater than 1, no triangle exists for those inputs — the answer is "no solution," not a rounded number. Treat sin B > 1 as a signal that the given side is too short.

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Frequently Asked Questions

Can I use the law of sines on a right triangle?

Yes. The law of sines holds for every triangle, including right triangles, because sin 90° = 1 simply makes the hypotenuse its own ratio. That said, for a right triangle the basic SOH-CAH-TOA trig ratios or the Pythagorean theorem calculator are usually quicker.
When should I use the law of sines vs. the law of cosines?

Use the law of sines when you have a side paired with its opposite angle (the AAS, ASA, and SSA cases). Use the law of cosines when you have two sides and the included angle (SAS) or all three sides (SSS), where no such pair exists to start from.
How do I find an unknown side using the law of sines?

Set up the ratio a/sin A = b/sin B using one known side–angle pair and the angle opposite the side you want. Solve by cross-multiplying: the missing side equals the known side times the sine of its opposite angle, divided by the sine of the known angle.
How do I find an unknown angle using the law of sines?

Use the reciprocal form sin A/a = sin B/b. Rearrange to sin B = b · sin A / a, compute the value, then take the inverse sine to get B. Remember to check whether 180° − B is also a valid second solution in the ambiguous SSA case.
What is the ambiguous case, and how do I know if there are two triangles?

The ambiguous (SSA) case is when two sides and a non-included angle can describe 0, 1, or 2 triangles. Compute the height h = b · sin A. For an acute angle A, there are two triangles when h < a < b, one when a = h or a ≥ b, and none when a < h. If A is 90° or more, there is one triangle when a > b and none otherwise.
Why does the law of sines work?

Drop an altitude h from one vertex of the triangle. It equals both b · sin A and a · sin B, so b · sin A = a · sin B, which rearranges to a/sin A = b/sin B. Repeating with another altitude extends the equality to the third side, giving the full rule.