The Law of Sines (Sine Rule)

The law of sines is used to finding missing sides and angles of triangles.


This formula can be used for triangles in the form of AAS, ASA, and SSA. Triangles in the form SSS and SAS require the law of cosines. These triangle names were first introduced when proving triangle congruence in geometry.

If you already know how to use the sin rule, jump straight to examples.

If not, view the rule and our explanation below:

law of sines    

Before you attempt to use the formula shown above, please be sure that you understand the basics of trigonometry first. You must also have strong algebra skills, particularly with solving proportions. Before going any further, please be certain you are able to:

  • Evaluate the sine of angle using the sin function
  • Find the measure of an angle using the inverse sine function: sin-1
  • Solve a proportion involving trig functions.
  • Understand the naming conventions for triangles (see below)
Naming Conventions for Sides and Angles of a Triangle

First, you must understand what the letters a, b, c and A, B, C represent in the formula. The corresponding letters (a & A, b & B, c & C) represent the side and the angle across from it. Please see the figure below.

Note: the letters used are arbitrary. As long as you match the side with the angle across from it, the formula will work!

The Law of Sines (also known as the Sine Rule)

The formula is listed in two forms: the standard form (used for sides) and the reciprocal form (used for angles). See below.

Frequently Asked Questions about the Sine Rule

Q: Why are there 3 parts to the equation?

A: The formula states that all 3 ratios are equal. You will never use all 3 of them at the same time. Instead, you will pick 2 of the 3, depending on what information you are given. The formula could be written as...

Q: Why are there two different forms of the law of sines?

A: The formula used for angles is known as the reciprocal form. In fact, it is not needed. We simply use it to make the algebra easier when we are solving problems to find a missing angle.

Q: What is the "Ambiguous Case?"

A: Please read below.

The Ambiguous Case of the Sin Rule

A special case of the law of sines arises when the triangle is in the form Side-Side-Angle (SSA). This is easy to remember because SSA is a curse word, and you will be cursing at your paper whenever you have to deal with it.

When you are given angle A, side a, and side b, it is possible that two different triangles can be made to fit the conditions.

  • Two triangles are formed when h < a < b, where h = bsinA.

Notice in the figure above that side a can "swing" to form either an obtuse triangle or an acute triangle, but angle angle A and side b are not change.

The figure at the right demonstrate the two possible cases. Case 1 is the entire triangle. Case 2 is the grey shaded triangle.  

Examples Using the Law of Sines
Example to find an Angle   Example to find a Side
Additional Information

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