Law of Cosines
The Law of Cosines
The Law of Cosines is a topic in Right Triangles & Trigonometry in the Common Core State Standards. It is aligned to Standard G-SRT.10, which requires students to prove the Law of Cosines and use it to solve problems involving general triangles.
The Law of Cosines \(c^2=a^2+b^2-2ab\cos C\) solves a triangle given two sides and the included angle (SAS) or all three sides (SSS).
Theory
The Law of Cosines relates all three sides of a triangle to one angle:
where \(C\) is the angle included between sides \(a\) and \(b\). Use it for:
- SAS — two sides and the included angle \(\Rightarrow\) the third side;
- SSS — all three sides \(\Rightarrow\) any angle.
The two forms:
How to use the Law of Cosines
- SAS: put the known angle as \(C\), its opposite side as \(c\), and solve for \(c\).
- SSS: use the rearranged form for \(\cos C\), then \(\cos^{-1}\).
- Finish the triangle with the Law of Sines if needed.
Apply the Law of Cosines.
| \(c^2\) | \(=\) | \(7^2+10^2-2(7)(10)\cos 60^\circ\) |
| \(=\) | \(149-140(0.5)=79\) | |
| \(c\) | \(=\) | \(\sqrt{79}\approx 8.9\) |
Rearrange for \(\cos C\).
| \(\cos C\) | \(=\) | \(\dfrac{5^2+6^2-7^2}{2(5)(6)}=\dfrac{12}{60}=0.2\) |
| \(C\) | \(=\) | \(\cos^{-1}(0.2)\approx 78.5^\circ\) |
SAS with the \(70^\circ\) angle between the two distances.
| \(d^2\) | \(=\) | \(12^2+9^2-2(12)(9)\cos 70^\circ\approx 151.1\) |
| \(d\) | \(\approx\) | \(12.3\ \text{mi}\) |
Common pitfalls
Frequently asked questions
What is the Law of Cosines?
\(c^2=a^2+b^2-2ab\cos C\), relating all three sides of a triangle to one angle.
When do you use the Law of Cosines?
For SAS (two sides and the included angle) or SSS (all three sides).
How does it relate to the Pythagorean theorem?
When the included angle is \(90^\circ\), the cosine term is zero and it becomes \(c^2=a^2+b^2\).
How do you find an angle with the Law of Cosines?
Rearrange to \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) and take the inverse cosine.