In any triangle, the ratio of a side length to the sine of its opposite angle

is the same for all three sides. As a formula:

is the same for all three sides. As a formula:

Try this
Drag any vertex of the triangle. Note that the ratio of a side to the sine of its opposite angle is the same for all three sides.

See also Law of Cosines

The Law of Sines says that in any given triangle, the ratio of any side length to the sine of its opposite angle is the same
for all three sides of the triangle.
This is true for *any* triangle, not just right triangles.

Press 'reset' in the diagram above.
Note that side 'a' has a length of 25.1, and its opposite angle A is 67°.
The sine of 67° is 0.921, so the ratio of 25.1 to 0.921 is 27.27.
If you repeat this for the other three sides, you will find they have the same ratio,
designated here by the letter *s*.

As you drag the above triangle around, you will see that although this ratio varies, it is always the same for all three sides of the triangle.

The Law of Sines is written formally as where A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side c.

A triangle has three sides and three angles. The Law of Sines is one of the tools that allows us to solve the triangle. That is, given some of these six measures we can find the rest. Depending on what you are given to start, you may need to use this tool in combination with others to completely solve the triangle.

You can use the Law of Sines if you already know

- One side and its opposite angle, and
- One or more other sides or angles

We now know both angles B and C, so using the fact that the interior angles of a triangle add up to 180°, we can find the third angle A:

Using the same principle as above we know that so we solve this for- click "hide details' then reshape the triangle by dragging its vertices.
- Solve the triangle using the Law of Sines.

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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