Key takeaways
- The sine rule says a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C) in any triangle.
- Each side must be paired with the angle directly opposite it.
- Use it when you know two angles and one side, or two sides and an angle opposite one of them.
- Two sides and a non-included angle (SSA) can give two different triangles.
- Make sure your calculator is in degree mode.
In this guide
What is the sine rule?
The sine rule, also called the Law of Sines, links the sides of any triangle to the angles opposite them. It works for every triangle, not just right-angled ones.
The idea is simple. The bigger an angle, the longer the side across from it. The sine rule turns that idea into an exact equation.
Why does the sine rule matter?
Many real triangles have no right angle. A plot of land, the path of a ship, the line from two phone towers to a caller. The basic SOH-CAH-TOA rules only work for right-angled triangles, but the sine rule works for all of them.
Measure one side and two angles, and the sine rule gives you every other side. You never have to reach the far corner.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a, b, c | The three side lengths | a = 10 m |
| A, B, C | The angles opposite those sides, in degrees | A = 30° |
| sin | The sine function on your calculator | sin(30°) = 0.5 |
To find a side, rearrange to b = a × sin(B) ÷ sin(A). To find an angle, flip the fractions: sin(B) ÷ b = sin(A) ÷ a, then use inverse sine.
When to use it
| What you know | Short name | Use |
|---|---|---|
| Two angles and any side | AAS or ASA | Sine rule |
| Two sides and an angle opposite one of them | SSA | Sine rule (watch for two answers) |
| Two sides and the angle between them | SAS | Cosine rule |
| All three sides | SSS | Cosine rule |
How to use the sine rule step by step
- 1Sketch and label the triangle. Put each small letter opposite its capital letter.
- 2Find a complete pair. You need one side and its opposite angle, like a and A.
- 3Write the rule with the pair you know and the thing you want:
a ÷ sin(A) = b ÷ sin(B). - 4Rearrange to get the unknown alone:
b = a × sin(B) ÷ sin(A). - 5Calculate with your calculator in degree mode.
- 6Use angles adding to 180° to find the third angle if you need it.
Try it yourself. Enter side a and angles A and B in the calculator below to find side b with every step shown.
Try it yourself
Pre-filled with the example — change any value.
This calculator finds side b from a, A and B. For the third side, work out angle C first (180° minus the other two) and use the same rule again with C in place of B.
The ambiguous case (SSA)
When you know two sides and an angle that is not between them, there can be two possible triangles. This is called the ambiguous case.
To check, add the second angle to the known angle. If the total is less than 180°, the second triangle exists too.
Common mistakes to avoid
- Calculator in radians.
sin(30)in radian mode is about −0.99, not 0.5. Check for a small D or DEG on the screen. - Missing the second triangle in the SSA case.
- Using the sine rule for SAS or SSS. Without a matching pair, you cannot start. Use the cosine rule instead.
- Rounding too early. Keep the full sine values until the last step.
Tips and tricks
- Right-angle shortcut: if C = 90°,
sin(C) = 1and the rule turns into ordinary SOH-CAH-TOA. - Stuck without a pair? Use the cosine rule calculator first, then switch to the sine rule.
- Need the area? With two sides and the angle between them, use the SAS triangle area calculator.
- Just need a sine value? The sin, cos, tan calculator gives it with steps.
Where you'll use it in real life
- Surveying: finding distances across rivers, valleys or private land.
- Navigation: working out how far a ship or plane is from two known points.
- Astronomy: estimating distances using angles measured from two places.
- Construction: finding roof rafter lengths when the roof angles are known.
Quick summary
The sine rule a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C) solves any triangle once you know a side and its opposite angle plus one more fact. Use it for AAS, ASA and SSA, check for a second triangle in the SSA case, and switch to the cosine rule for SAS or SSS. The calculator above shows every step.
Frequently asked questions
When should I use the sine rule vs the cosine rule?
Use the sine rule when you know two angles and a side, or two sides and an angle opposite one of them. Use the cosine rule when you know all three sides, or two sides and the angle between them.
What is the ambiguous case?
It happens when you know two sides and an angle that is not between them (SSA). There may be two possible triangles, one, or none, because two different angles can have the same sine.
Does the sine rule work for right-angled triangles?
Yes. It works for every triangle. For a right-angled triangle, sin(90°) = 1, so it becomes the usual SOH-CAH-TOA rules.
Why do I get a strange answer on my calculator?
Your calculator is probably in radian mode. Switch it to degree mode so that sin(30) gives 0.5.
