Law of Sines and Law of Cosines

When SOH CAH TOA stops working

SOH CAH TOA only works for right triangles. For non-right triangles, you need two more rules.

Labeling convention: side $a$ is opposite angle $A$, side $b$ opposite $B$, side $c$ opposite $C$.

Law of Sines

$\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$

Use when you know:

• Two angles and any side (AAS or ASA), or
• Two sides and an angle opposite one of them (SSA — the "ambiguous case").

Example (AAS). $A = 50^\circ$, $B = 60^\circ$, $a = 8$. Find $b$.

$\dfrac{\sin 50^\circ}{8} = \dfrac{\sin 60^\circ}{b}$ $b = \dfrac{8 \sin 60^\circ}{\sin 50^\circ} \approx \dfrac{8 \cdot 0.866}{0.766} \approx 9.05$

Law of Cosines

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

(Similar formulas for $a^2$ and $b^2$ by cycling letters.) Notice if $C = 90^\circ$, $\cos 90^\circ = 0$ and this becomes the Pythagorean theorem.

Use when you know:

• Two sides and the angle between them (SAS), or
• All three sides (SSS).

Example (SAS). $a = 5$, $b = 7$, angle between them $C = 40^\circ$. Find $c$.

$c^2 = 25 + 49 - 2 \cdot 5 \cdot 7 \cdot \cos 40^\circ$ $c^2 = 74 - 70 \cdot 0.766 \approx 20.4 \implies c \approx 4.52$

Example (SSS). $a = 4$, $b = 5$, $c = 6$. Find angle $C$.

$36 = 16 + 25 - 2 \cdot 4 \cdot 5 \cos C$ $-5 = -40 \cos C \implies \cos C = 0.125 \implies C \approx 82.8^\circ$

The ambiguous case (SSA)

Two sides and an angle opposite one of them sometimes gives two valid triangles, one triangle, or no triangle. Sketch the picture — if the "swing" of the opposite side can reach the base in two places, both are valid.

Quick chooser

Known	Use
AAS, ASA	Law of Sines
SSA	Law of Sines (ambiguous case)
SAS	Law of Cosines
SSS	Law of Cosines

Key takeaways

• Law of Sines: ratios of $\sin(\text{angle})$ to the opposite side are equal.
• Law of Cosines: like Pythagorean theorem, with a $-2ab\cos C$ correction.
• SAS or SSS → Law of Cosines. Otherwise → Law of Sines.