What is the Law of Sines? (Sine Rule Explained)

The law of sines, also known as the sine rule, is one of the most essential formulas in trigonometry. But what is the law of sines? It states that in any triangle, the ratio between the length of a side and the sine of the opposite angle is constant. In other words, the sides and angles of a triangle are directly related by simple proportions.

If you want to try it directly, you can use our law of sines calculator 🇺🇸 to compute sides and angles in just a few clicks.

The law of sines formula is beneficial for solving non-right triangles, for which the Pythagorean theorem and elementary trigonometric ratios are insufficient. Take a look at the law of sines formula to understand this principle better:

We can take the inverse, so the law of sines can also be written as follows:

In other words, the law of sines states that the ratio between the sine of an angle and the length of the opposite side is the same for all angles in a triangle. Applying this formula lets you easily determine the missing sides or angles when given an angle–side pair.

However, in some instances, this rule can lead to more than one solution. This is known as the ambiguous case of the sine rule, which occurs in side-side-angle (SSA) configurations.

Knowing the answer to "What is the law of sines?" is one thing, but knowing its proof is also a good idea. Indeed, the reasoning in evidence allows us to apply that same reasoning to similar problems.

Let's consider the triangle $ABC$ with sides $a$, $b$, $c$ and opposite angles $\alpha$, $\beta$, $\gamma$.

First, draw an altitude $h$ from vertex $B$ (angle $\beta$) to the base $b$, splitting it into two segments $b_1$ and $b_2$.

In the left right triangle (with hypotenuse $a$ and angle $\gamma$), this gives us:

Similarly, in the right right triangle (with hypotenuse $c$ and angle $\alpha$), we get:

Then, equate the two expressions for $h$, which gives us:

Repeat the same argument for a different altitude (for example, from vertex $A$). We, then, get:

Finally, combine the two results to obtain the law of sines:

If you want to explore more trigonometric properties, try our trigonometry calculator 🇺🇸.

You can use the law of sines 🇺🇸 to find the lengths of sides or unknown angle measurements in any type of triangle. In particular, you should use the sine rule when you know:

• Two sides and one angle opposite one of them (SSA), in order to find the measure of an unknown angle; or
• Two angles and one side (ASA or AAS) if you want to find an unknown length.

On the other hand, the law of cosines is more appropriate when you have:

• Three sides (SSS), and want to calculate one of the angles; or
• Two sides and the included angle (SAS) in order to find the third side.

If you don't know which one to apply, check out our comprehensive guide: Law of Sines vs. Law of Cosines: What's the Difference?.

Let's take a triangle in which one angle is $\alpha = 80^\circ$ with opposite side $a$ and another angle is $\beta = 30^\circ$ with opposite side of length $b = 12\ \text{in}$.

We want to determine the length $a$.

1. Apply the law of sines:

2. Solve for $a$:

3. Finally, approximate the result:

Thus, the side opposite the $80^\circ$ angle has length $a \approx 23.6 \, \text{in}$.

After reading this article, we hope you will confidently answer when asked "what is the law of sines?". Roughly, you should say something like: the law of sines establishes a relationship between the sides of a triangle and the sines of its opposite angles, helping people find solutions to problems in triangles. Moreover, using the law of sines formula makes it easy to determine the missing sides or angles when a pair of angles and sides is known.