Triangle : Law Of Sines & Cosines
What does ‘solving the triangle’ mean?
It means that if we are given some facts about a triangle, we can find some or all of the rest. For example, if we know two sides of a right triangle we can find (or ‘solve for’) the third side using Pythagoras’ Theorem.
To completely solve a triangle it usually means finding everything about it – all three sides and all three angles. But often we are just interested in one unknown aspect of the triangle.
Tools for solving triangles
| Tool | Given | You can find |
| Interior angles sum to 180° | Any two interior angles | The third angle |
| Pythagoras’ Theorem | Two sides of a right triangle | The third side |
If you are familiar with trigonometry, the following tools can be used with any triangle:
| Tool | Given | You can find |
| Law of Cosines | Two sides and the angle between them | The third side |
| Law of Sines | One side and its opposite angle, and one other item | Everything else |
Again, if you are familiar with trigonometry, the following tools can be used with right triangles:
| Tool | Usage |
| Sine | These trigonometry functions relate an angle and two other sides. If you know two of the three, you can find the other. It is beyond the scope of the this volume on Geometry to define them here. See the Trigonometry Overview for a brief description. |
| Cosine | |
| Tangent |
The tools listed above can be used in any combination. There is no ‘right way’ of solving a triangle (or any other geometric problem). Just as in woodworking, you can use the tools in many ways and in many sequences. Think about what you have been given to start, and find tools that lead you in the direction you want to go.
Law of Sines
is the same for all three sides. As a formula:
The Law of Sines says that in any given triangle, the ratio of any side length to the sine of its opposite angle is the same for all three sides of the triangle. This is true for any triangle, not just right triangles.
Press ‘reset’ in the diagram above. Note that side ‘a’ has a length of 25.1, and its opposite angle A is 67°. The sine of 67° is 0.921, so the ratio of 25.1 to 0.921 is 27.27. If you repeat this for the other three sides, you will find they have the same ratio, designated here by the letter s.
As you drag the above triangle around, you will see that although this ratio varies, it is always the same for all three sides of the triangle.
Written as a formula
The Law of Sines is written formally as
where A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side c.
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What is it used for?
A triangle has three sides and three angles. The Law of Sines is one of the tools that allows us to solve the triangle. That is, given some of these six measures we can find the rest. Depending on what you are given to start, you may need to use this tool in combination with others to completely solve the triangle.
When do I use it?
You can use the Law of Sines if you already know
- One side and its opposite angle, and
- One or more other sides or angles
The first allows us to calculate the “Law of Sines” ratio s. Then we can use this ratio to find other sides and angles using the other givens.
The circumcircle connection
It turns out that the “Law of Sines” ratio is also the diameter of the triangle’s circumcircle, which is the circle that passes through all three vertices of the triangle. This is sometimes formally written as
where r is the circumradius – the radius of the triangle’s circumcircle.
So if we are given one side and its opposite angle we can find the “law of Sines” ratio for the triangle. Then, using that ratio and the other given elements, we can solve the triangle.
Law of Cosines
the Law of Cosines formula:
The Law of Cosines is a tool for solving triangles. That is, given some information about the triangle we can find more. In this case the tool is useful when you know two sides and their included angle. From that, you can use the Law of Cosines to find the third side. It works on any triangle, not just right triangles.
The Law of Cosines is written formally as
| c2 = a2 + b2 – 2ab cos(C) |  Calculator |
where a and b are the two given sides, C is their included angle, and c is the unknown third side. See figure above.
To illustrate, press ‘reset’ in the diagram above. Note that side a has a length of 30, and side b has a length of 18.9. Their included angle C is 58°. By plugging these into the Law of Cosines we get a length of 25.6 for the the third side c.
As you drag the above triangle around, this calculation will be updated continuously to show the length of the side c using this method.
Example
We are given a triangle with two sides (a,b) and the included angle C, as shown below.
We will find the third side.
- We start with the formula:
c2=a2+b2−2abcosC
- Insert the values for a,b and C:
c2=23.62+8.22−2·23.6·8.2·cos(60°)
- Evaluate the right side:
c2=430.6
- Finally we get
c=√430.68=20.8
Giving us the length of the third side c.
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