Solving Triangles Using the Law of Cosines
Rule

Law of Cosines

Consider with sides of length and which are respectively opposite the angles with measures and
triangle with angles and sides labeled

The following equations hold true with regard to

Proof

For Acute Angles
The first equation will be proven. The other two equations can be proven by following the same procedure.
Begin by drawing the altitude from to its opposite side
acute triangle and its height
By the definition of an altitude, both and are right triangles. By applying the Pythagorean Theorem to and two equations can be obtained.
The binomial in Equation I can be expanded.
Notice that Equation II says that is equal to Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.
Now, the term in this equation can be written using the cosine ratio.
acute angle of the triangle marker
In the cosine of is the ratio of to
Finally, can be substituted for into By doing so, the formula for the Law of Cosines is obtained.

Proof

For Obtuse Angles
The first equation will be proven for obtuse angles. The remaining equations can be proven similarly.
Consider with side lengths of and respectively opposite the angles with measures and such that is greater than
obtuse triangle

The altitude of the triangle is the perpendicular segment from to the extension of the base Let be the endpoint of this segment and be the distance from to

obtuse triangle and its height
From the definition of an altitude, it follows that and are right triangles. Two equations can be obtained by applying the Pythagorean Theorem to both triangles.
Expand the binomial in Equation I.
From Equation II, is equal to Therefore, the expanded form of Equation I can be rewritten by using the Substitution Property of Equality.
Note that and are supplementary angles. Using the cosine ratio of then gives an expression for the term.
obtuse triangle and angle A prime marked
In the cosine of is the ratio of to
By the Sine and Cosine of Supplementary Values Angles, and have opposite values.
Finally, by substituting for into the Law of Cosines is obtained.
Exercises