Overview

Use the Law of Cosines when a triangle is not right but you need a missing side or angle.

Key Ideas

• $a^2 = b^2 + c^2 - 2bc\cos A$ and cyclic versions.
• $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.
• If $A=90^\circ$, the formula reduces to the Pythagorean theorem.

Core Skills

Choose the Correct Opposite Side

Always match $a$ with $A$, $b$ with $B$, and $c$ with $C$. The side opposite the angle is the one being solved for in $a^2=b^2+c^2-2bc\cos A$.

Find a Side from Two Sides and an Angle

When two sides and their included angle are known, compute the third side directly using the first form.

Find an Angle from Three Sides

Use $\cos A = (b^2+c^2-a^2)/(2bc)$, then take $\arccos$ if needed.

Worked Example

Triangle has sides $a=7$, $b=8$, $c=9$. Find $\cos A$.

$\cos A = \frac{8^2 + 9^2 - 7^2}{2\cdot 8\cdot 9} = \frac{64+81-49}{144} = 2/3$.

More Examples

Example 1: Find a Side

Triangle has sides $b=6$, $c=10$, and included angle $A=60^\circ$. Find $a$.

$a^2 = 6^2 + 10^2 - 2\cdot 6\cdot 10\cdot \cos 60^\circ = 36+100-60=76$, so $a=\sqrt{76}$.

Example 2: Detect an Obtuse Angle

Triangle has sides $a=9$, $b=5$, $c=7$. Is $\angle A$ obtuse?

Compute $b^2 + c^2 - a^2 = 25 + 49 - 81 = -7$, so $\cos A < 0$ and $A$ is obtuse.

Strategy Checklist

• Identify the included angle when two sides are known.
• Use the cosine form to check for obtuse or acute angles.
• Reduce to Pythagorean theorem when $A=90^\circ$.

Common Pitfalls

• Using the wrong opposite side for angle $A$.
• Dropping the minus sign on $2bc\cos A$.
• Forgetting to square all side lengths in the equation.

Practice Problems