Law of Cosines Calculator

The Law of Cosines generalizes the Pythagorean theorem to include all types of triangles, not just right-angled ones. For any triangle with sides labeled $a$, $b$, and $c$, and corresponding opposite angles $\alpha$, $\beta$, and $\gamma$, the fundamental relationship is expressed as $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$. To find a missing side when two sides and the included angle (SAS) are known, we take the square root of the rearranged formula: $c = \sqrt{a^2 + b^2 - 2ab \cos(\gamma)}$. Conversely, if all three sides (SSS) are known, the formula can be rearranged to solve for any angle, such as $\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}$.

Imagine you have a triangle in a Side-Angle-Side (SAS) configuration where side $a = 7$ units, side $b = 10$ units, and the included angle $\gamma = 60^\circ$. To find the missing side $c$, substitute these values into the Law of Cosines formula. First, square the known sides: $7^2 = 49$ and $10^2 = 100$. Next, calculate the product term: $2 \times 7 \times 10 \times \cos(60^\circ)$. Since $\cos(60^\circ) = 0.5$, this product equals $140 \times 0.5 = 70$. Now, combine the terms: $c^2 = 49 + 100 - 70$, which simplifies to $c^2 = 79$. Finally, take the square root to find $c = \sqrt{79} \approx 8.89$ units.

Always verify your calculator's angle mode setting (degrees versus radians) before starting any trigonometric calculations. When solving for angles using inverse cosine functions, double-check that your side lengths satisfy the triangle inequality theorem—the sum of any two sides must strictly exceed the remaining side. Keep intermediate decimal values unrounded until the final step to maintain high precision in engineering or architectural blueprints.