What Is the Law of Sines?

Triangle with the law of sines

The unit circle shows us that sin 1(A) can be one of two angles in the interval [0, 180]. So, it’s very important to make sure you’ve gotten the correct angle when using the law of sines.

If the angle you find seems wrong in relation to the figure or the information you’ve been given, try to find 180 minus the angle you found, like this:

180 v

Note! It’s often clever to put the unknown in the top left of these equations. Remember that you only use two of the three terms for any problem! The formula has three terms in it to show that they’re all equal, and that you can use any pair of them.

Formula

The Law of Sines

Given two angles and one side, or two sides and one angle of a triangle ABC, then

a sin A = b sin B = c sin C (1) sin A a = sin B b = sin C c (2)

Rule

Uses

You can use the law of sines to

  • Find a side, if you know two angles and a side that is opposite to one of the known angles—Formula (1).

  • Find an angle, if you know one angle and two sides, one of which is opposite to the angle you want to find—Formula (2).

Example 1

You have a quadrilateral ABCD with AB = 12, AD = 6, CD = 5 and ABD = 30. Find A and the diagonal BD.

It’s useful to draw an auxiliary figure. It will look like this:

Example of using the law of sines

Begin by finding the angle ADB, to find A:

sin ADB 12 = sin 30 6 | 12 sin ADB = 12 sin 30 6 = 1 ADB = sin 1(1) = 90 A = 180 90 30 = 60

Then, to find the diagonal BD, you can either use the Pythagorean Theorem or the law of sines. Let’s look at how to solve it with the law of sines:

BD sin 60 = 6 sin 30 | sin 60 BD = 6 sin 60 sin 30 6 0.866 0.5 10.4

Hence, the diagonal BD 10.4.

Example 2

A triangle ABC is defined by A = 40, AC = 8.0cm and BC = 6.0cm. Draw the triangle, and find the sizes of the remaining sides and angles.

Start by drawing a line segment l and mark a point A. Construct an angle A = 40. Then mark AC = 8.0cm along the left side of A and call that point C. You still don’t know the position of B, but you know that BC = 6.0cm, so set this as the distance between the legs of your draft compass. Set the point of the draft compass on C and make an arc that intersects l in two points. Call these points B1 and B2. This means that there are two triangles that meet the criteria, as you can see in the figure below.

Example of using the law of sines 2

Look at AB1C first.

Example of using the law of sines 3

Find B by using the law of sines:

sin B1 8 = sin 40 6 | 8 sin B1 0.857 B1 59

This implies that C is

C 180 59 40 = 81

Then you can find the side AB1 with the law of cosines:

AB12 = 62 + 82 2 8 6 cos 81 = 85 AB1 = 85 9.22

Then you can look at AB2C.

Example of using the law of sines 4

Begin with the angle B2. You find this angle through theory about supplementary angles. From the figure above, you can see that B1B2C is an isosceles triangle. This means that B1 and B2 are supplementary angles, such that

B2 = 180 B 1 180 59 121

B2 = 180 B 1 180 59 = 121

For that reason,

C = 180 B 1 B2 180 40 121 19

C = 180 B 1 B2 180 40 121 = 19

Finally, you can find AB2 through the law of sines:

AB2 sin 19 6 sin 40 | sin 19 AB2 3

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