It is most useful for solving for missing information in a triangle. For example, if all three sides of the triangle are known, the cosine rule allows one to find any of the angle measures. Similarly, if two sides and the angle between them is known, the cosine rule allows one to find the third side length.
Given the following triangle with corresponding sides length , , and
the law of cosines states that
It can be seen as a generalization of the Pythagorean theorem. Take one arbitrary side of the triangle, for instance, . Then its square equals the sum of the squares of the other two sides, i.e. . If you were to plug in 90 degrees, you would be left with the Pythagorean theorem. Since the angle that faces our arbitrary side is not necessarily , we will have to subtract something, as the identity does not hold yet. The right side of this equation is still "too big." That something we have to subtract becomes .
We'll prove for side . Let's denote its facing angle as . The other two equations can be done in a similar way.
By definition we have Using the Pythagorean theorem, we get Substituting for and , we get
- The identity is also known as the Pythagorean identity.
- This proof isn't perfect. We should have been worried about angles. This can be avoided by using directed angles.
Let denote the dot product between and .
Also, let , , and .
In triangle , we have and . Determine .
Applying the cosine rule on , we get
Note: This is similar to the 'ambiguous case' of sine rule, since we have , which is the condition .
In triangle , and Find the measure of
Applying the law of cosines on side we get
Try the following problem:
When it comes to the applications with regards to the law of cosines, it has its own applications towards vector quantities (and not qualities).
- Triangle Inequality
- Pythagorean Theorem
From the cosine rule, we have
and by taking the square root of both sides, we have , which is also known as the triangle inequality. One useful application of the triangle inequality is to test if three given lengths can define a triangle.
The Pythagorean theorem applies to right triangles, so let be a right angle, i.e., . Then by the cosine rule,
Suppose and are positive reals such that and . Show that .
Since , it follows that and we have similar inequalities for other variables. Hence, the numbers and satisfy the triangle inequality, and there exists a triangle such that . As such
which gives us that . Similarily, we have and , which show that triangle is equilaterial, implying that .
Note: This may also be done directly by summing up the 3 equations to get .
One big and one small circle share the same center .
Then is constructed such that points and are on the big circle while point is on the smaller circle. intersects the smaller circle at and and passes through and , as shown above.
If (red segment), (blue segment), and (green side), what is the radius of the smaller, orange circle?