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Law of Sines Calculator

Table of contents

Law of sines formulaLaw of sines applicationLaw of sines calculator – how to use it?FAQs

This law of sines calculator is a handy tool for solving problems that include lengths of sides or angles of a triangle. We will explain the law of sines formula and give you a list of cases in which this rule can be deemed useful. Thanks to this triangle calculator, you will now be able to solve some trigonometry problems (more elaborate than using the Pythagorean theorem).

However, if you don't know what the sine is, first check out our sine calculator.

Prefer watching over reading? Learn all you need in 90 seconds with this video we made for you:

Law of sines formula

illustration of law of sines. Triangle with sides a, b, c and angles α, β, γ.

The law of sines states that the proportion between the length of a side of a triangle to the sine of the opposite angle is equal for each side:

a / sin(α) = b / sin(β) = c / sin(γ)

This ratio is also equal to the diameter of the triangle's circumcircle (circle circumscribed on this triangle).

Note that you can use this law for any triangle and one that isn't a right triangle. If you are only interested in solving problems related to right triangles, our right triangle calculator might be more useful for you.

Law of sines application

You can transform the law of sines formulas to solve some problems of triangulation (solving a triangle). You can use them to find:

  1. The remaining sides of a triangle, knowing two angles and one side.
  2. The third side of a triangle, knowing two sides and one of the non-enclosed angles. In some cases (ambiguous cases), there may be two solutions to the same triangle. If the following conditions are fulfilled, your triangle may be an ambiguous case:
    • You only know the angle α and sides a and c;
    • Angle α is acute (α < 90°);
    • a is shorter than c (a < c); and
    • a is longer than the altitude h from angle β, where h = c × sin(α) (or a > c × sin(α)).

You can also combine these equations with the law of cosines to solve all other problems involving triangles.

Law of sines calculator – how to use it?

  1. Start with formulating your problem. For example, you may know two angles and one side of the triangle and be looking for the remaining sides.
  2. Input the known values into the appropriate boxes of this triangle calculator. Remember to double-check with the figure above whether you denoted the sides and angles with the correct symbols.
  3. Watch our law of sines calculator perform all calculations for you!

Can I use the law of sines on right triangles?

Yes, the law of sines works for all triangles. To use it, you need to know either two sides and an angle opposite to one of these sides or two angles and one side of the triangle.

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

Use the law of sines when you know either:

  • Two angles and one side; or
  • Two sides and an angle opposite to one of these angles.

Use the law of cosines when you know either:

  • Three sides; or
  • Two sides and the angle between them.

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

To find side a given side b and the angles α and β that are opposite to a and b, respectively, we apply the law of sines a / sin(α) = b / sin(β).

Solving for a we arrive at a = b × sin(α) / sin(β).

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

To find the angle α given the side a opposite to α as well as the side b and its opposite angle β, we apply the formula derived from the law of sines: sin(α) = a × sin(β) / b, which we can further transform to α = arcsin(a × sin(β) / b), where arcsin is the arcsine function.

How do I apply the law of sines to 30 60 90 triangle?

Let's say a is the side opposite to angle 30°, b to angle 60°, and c to 90°.

  1. The law of sines says that a / sin(30°) = b / sin(60°) = c / sin(90°).
  2. Plugging in the values of sines, we obtain 2a = 2b/√3 = c.
  3. Now, you can express each of a,b,c with the help of any other of them. For instance, b and c expressed with the help of a read: c = 2 × a and b = √3 × a.
Illustration of law of sines. Triangle with sides a,b,c and angles α β γ.
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