The Law of Cosines relates the three sides of a triangle to one angle. It is useful when the
Law of Sines is not enough, especially in SAS and SSS triangle cases.
\[
a^2=b^2+c^2-2bc\cos A
\]
\[
b^2=a^2+c^2-2ac\cos B
\]
\[
c^2=a^2+b^2-2ab\cos C
\]
1. When to use the Law of Cosines
Use the Law of Cosines when you know:
- two sides and the included angle, called SAS;
- all three sides, called SSS.
In standard notation, side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\),
and side \(c\) is opposite angle \(C\).
2. SAS: finding the third side
If sides \(a\) and \(b\) and included angle \(C\) are known, then:
\[
c^2=a^2+b^2-2ab\cos C
\]
After finding \(c\), the remaining angles can be found using the Law of Cosines or the Law of Sines.
3. SSS: finding an angle
If all three sides are known, rearrange the Law of Cosines:
\[
\cos C=\frac{a^2+b^2-c^2}{2ab}
\]
\[
C=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)
\]
Similar formulas find \(A\) and \(B\).
4. Example
Suppose \(a=5\), \(b=7\), and \(C=60^\circ\). Then:
\[
c^2=5^2+7^2-2(5)(7)\cos60^\circ
\]
\[
c^2=25+49-70\cdot\frac12
\]
\[
c^2=39
\]
\[
c=\sqrt{39}
\]
5. Formula summary
| Goal |
Formula |
Use case |
| Find side \(a\) |
\(a^2=b^2+c^2-2bc\cos A\) |
Known \(b\), \(c\), and included angle \(A\) |
| Find side \(b\) |
\(b^2=a^2+c^2-2ac\cos B\) |
Known \(a\), \(c\), and included angle \(B\) |
| Find side \(c\) |
\(c^2=a^2+b^2-2ab\cos C\) |
Known \(a\), \(b\), and included angle \(C\) |
| Find angle \(C\) |
\(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) |
Known all three sides |
6. Triangle inequality
For three side lengths to form a triangle, each pair of sides must add to more than the third side:
\[
a+b>c,\qquad a+c>b,\qquad b+c>a
\]
7. Common mistakes
- Use the included angle for SAS. The included angle is between the two known sides.
- Do not forget the minus sign in \(a^2=b^2+c^2-2bc\cos A\).
- Check that side lengths satisfy the triangle inequality.
- When finding angles, make sure the value inside \(\arccos\) is between \(-1\) and \(1\).
- Keep angle units consistent: degrees with degrees, radians with radians.
