The Sine Rule

The sine rule, also known as the law of sines, is a mathematical relationship that applies to triangles. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.

That is...

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the opposite angles.

The sine rule is useful for several reasons:

1. Finding unknown side lengths: The sine rule can be used to find the length of an unknown side of a triangle if the lengths of the other two sides and the opposite angle are known.

2. Solving triangles: The sine rule can be used to solve triangles, which means finding all the angles and side lengths of a triangle given some of the measurements.

3. Trigonometry: The sine rule is a fundamental tool in trigonometry, which is the branch of mathematics that deals with the relationships between the sides and angles of triangles.

4. Navigation: The sine rule is used in navigation, for example, in calculating distances between two points on the earth's surface or determining the height of a lighthouse.

We can use two versions of the sine rule...

To find a side of a triangle, place the sides on top:

To find an angle in a triangle, place the angles on top:

Example

Use the sine rule to find the length of BC

We have information about two angles and a side. Length BC = a and we can solve as follows:

Example

Use the sine rule to find the angle B

Notice the use of sin-1 - this is because we divide both sides by sine to isolate the angle (1/sin = sin-1). Make sure that you hold values in your calculator until the final solution - only then should you round the value.