Law of Cosines

Decide by the given pieces. The law of cosines is responsible when no complete side-opposite-angle pair is available: in the SAS case (two sides and the included angle, third side sought) and in the SSS case (three sides, angles sought). The law of sines, by contrast, always needs a pair of a side and its opposite angle and thus serves $\frac{ASA}{AAS}$ and SSA. Practical exam procedure: sketch, mark the given pieces, check whether a side-opposite-angle pair is complete. If yes: law of sines (shorter). If no: law of cosines. Often you combine both, for example first computing the third side from SAS with the law of cosines and then finding the remaining angles conveniently with the law of sines or the law of cosines again.

Solve c² = a² + b² − 2ab·cos γ for cos γ: move $c^{2}$ and the cosine term to different sides, then cos γ = (a² + b² − c²)/(2ab), and $\gamma{} = \arccos{}$ of that. Example with $a = 4$, $b = 6$, $c = 8$: $\cos{} \gamma{} =\frac{ (16 + 36 - 64)}{48} = -\frac{12}{48} = -0.25$, so $\gamma{} = \arccos{}(-0.25) \approx{} 104.5$°. This works for every angle if you swap the roles cyclically: $\cos{} \alpha{} =\frac{ (b^{2} + c^{2} - a^{2})}{(2bc)}$. A big advantage over the law of sines: arccos gives a unique result in the range 0° to 180°, there is no ambiguity. The sign directly tells you the angle type: positive cosine means acute, negative obtuse.

The law of cosines is the generalization of Pythagoras to arbitrary triangles: c² = a² + b² − 2ab·cos γ. If you set $\gamma{} = 90$°, $\cos{} \gamma{} = 0$, the correction term vanishes and $c^{2} = a^{2} + b^{2}$ remains, the classical Pythagoras. For acute angles $(\cos{} \gamma{} > 0)$ something is subtracted, c is shorter than in the right-angled case; for obtuse angles $(\cos{} \gamma{} < 0)$ something is effectively added, c is longer. The term −2ab·cos γ measures exactly how far the triangle deviates from the right angle. This view also serves as a mental plausibility check: if the included angle is obtuse, the computed opposite side must be longer than $\sqrt{(a^{2} + b^{2})}$, otherwise something is wrong.

In a triangle all angles lie between 0° and 180°, and on this range the cosine is a unique, decreasing function: positive for acute angles (0° to 90°), zero at 90°, negative for obtuse angles (90° to 180°). So if rearranging the law of cosines yields a negative value like $\cos{} \gamma{} = -0.25$, you know immediately without a calculator: γ is obtuse, here $\arccos{}(-0.25) \approx{} 104.5$°. That is a real information gain over the law of sines, where $\sin{} \beta{} = 0.9$ does not reveal whether $\beta{} \approx{} 64$° or ≈ 116° is meant. Hence the recommendation: compute angles, especially the largest angle of a triangle (opposite the longest side), preferably with the law of cosines, where obtuseness is detected automatically.

In the SAS case: insert the two known sides and the INCLUDED angle and take the root at the end. Example: $b = 3$, $c = 5$ and $\alpha{} = 50$° (note: α lies opposite the sought side a and between b and c). $a^{2} = b^{2} + c^{2} - 2bc\cdot{}\cos{} \alpha{} = 9 + 25 - 30\cdot{}\cos{} 50$° = 34 − 30·0.643 ≈ 14.7, so $a \approx{} 3.84$. Three checks pay off: first, the inserted angle must really lie between the two known sides (and thus opposite the sought side). Second, verify the calculator is in DEG mode. Third, triangle inequality as plausibility: a must lie between $|b - c| = 2$ and $b + c = 8$, and 3.84 fits. Forgetting the final square root is the most banal yet most frequent error.