Law of Cosines

The law of cosines generalizes the Pythagorean theorem to any triangle and solves SAS or SSS information.

Side and angle information determine which triangle relationship applies. A non-right triangle has three labelled sides and angles, plus a dropped altitude. A right-angle marker and side-length labels connect trigonometric ratios, area, and Pythagorean reasoning.

The side isolated on the left must be opposite the angle used in the cosine term.

The law

For sides $a,b,c$ opposite angles $A,B,C$,

c^2=a^2+b^2-2ab\cos C.

Equivalent forms isolate $a^2$ with angle $A$ or $b^2$ with angle $B$. Relabel rather than forcing every question into one memorized letter arrangement.

Worked example: SAS

The result lies between the difference $3$ and sum $17$ of the other sides, satisfying the triangle inequality.

Finding an angle from SSS

Rearrange:

\cos C=\frac{a^2+b^2-c^2}{2ab},

then

C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right).

Use the side opposite the desired angle as $c$. The largest angle should lie opposite the largest side.

Solving a full triangle

After cosine law gives one side or angle, use angle sum, another cosine-law equation, or sine law. If using sine law to find an angle, consider whether ambiguity applies; knowing the largest angle from SSS can simplify this.

Keep unrounded values until the final reporting step.

Common mistakes

Using an angle not opposite the isolated side. Match the pair.

Using a non-included angle with two known sides as SAS. Identify the information type.

Forgetting the square root when finding a side. The formula gives $c^2$.

Entering $-2ab\cos C$ with missing parentheses. Preserve operation order.

Accepting a side that violates triangle inequality. Use geometry to check.

Quick self-check

Is the information SAS or SSS?
Which side lies opposite the chosen angle?
Does the formula match those labels?
Is calculator mode in degrees or radians as required?
Does the result respect largest side/angle and triangle inequality?
Were exact values and precision preserved until the end?