Law of Sines and Cosines: How to Choose the Right Law for Any Triangle

Every triangle carries six parts — three sides and three angles. Any three of them, provided one is a side, lock in the other three. That is the promise of the law of sines and cosines: measure what you can reach, compute what you cannot.

The canyon triangle holds exactly three knowns: sides $a = 30$ and $b = 54$, meeting at $C = 46°$. The third side stays unmeasured — for now.

Naming is standard: side $a$ faces angle $A$, side $b$ faces $B$, side $c$ faces $C$. Blue marks the knowns; the red question mark is the wanted side. Every law of sines and cosines problem starts from a picture like this. Which tool fits the job? Read the givens, and the next block turns them into the law of sines and cosines decision map.

Triangles arrive in four data shapes, and each shape names its own law of sines and cosines tool. A 1908 surveying classic listed them: two angles and one side; two sides and the included angle; two sides and an opposite angle; three sides. Modern classes code them ASA, SAS, SSA, SSS — and the law of sines and cosines splits them between two tools:

Three steps pick the tool in seconds:

1. List the givens. Count the sides and angles; write the case code.
2. Hunt a side–angle pair. A side with its own opposite angle lets the law of sines fire at once. SSS and SAS own no pair, so the law of cosines makes the entry.
3. SSA? Test the supplement. The law of sines and cosines sine half reports an acute angle; the partner $180° - B$ gets a trial too.

Four rows, two tools — the law of sines and cosines division of labor. Run the steps on the canyon data (SAS). The map below names the cosine law: the law of sines and cosines verdict in one glance.

Law of Sines — the sides of a triangle are proportional to the sines of the opposite angles. This half of the law of sines and cosines lives on ratios:

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

Each ratio carries a bonus meaning: any side over the sine of its angle equals $2R$, the circumscribed circle's diameter.

Law of Cosines — the square of any side equals the squares of the other two, minus twice their product times the cosine of the included angle. This half of the law of sines and cosines lives on squares:

$a^2=b^2+c^2-2bc\cos A$

Rotate the letters for $b^2$ and $c^2$; all three forms share one shape.

Two formulas, one toolkit — the law of sines and cosines pair. Why this pairing and never the reverse? A 1912 textbook states it best. An angle found through a sine may be ambiguous, since supplementary angles share one sine. An angle found through a cosine is never ambiguous, since each cosine matches exactly one angle. That is the deep reason SSS angle-hunting belongs to the cosine side of the law of sines and cosines toolkit. Ratios for pairs; squares for the rest.

Hand the sine half of the law of sines and cosines two sides and a $\sin^{-1}$ key confesses only the acute candidate. The obtuse partner, $180° - B$, carries the same sine and deserves a trial. A classic drill: $C = 60°$, $b = 2\sqrt{3}$, $c = 3\sqrt{2}$, find $B$.

$\sin B=\frac{b\sin C}{c}=\frac{2\sqrt{3}\cdot\frac{\sqrt{3}}{2}}{3\sqrt{2}}=\frac{1}{\sqrt{2}}$

So $B = 45°$ or $B = 135°$. Test the partner: $60° + 135° = 195° > 180°$, so $135°$ is out. Then $B = 45°$ and $A = 75°$. The supplement test is the last gate of every law of sines and cosines decision.

SSA counting in one breath. No triangle if the swinging side falls short of $b\sin A$. One triangle if the given angle is obtuse or the swinging side is longer than the other given side. Two only when the angle is acute and the swinging side falls between those two lengths. That is the wobbly row of the law of sines and cosines map — its full walkthrough is on the ambiguous-case page.

Problem. A triangular brace has sides $a = 7$, $b = 5$, $c = 8$. Find all three angles. (Hall & Knight, 1912)

Answer: $A = 60°$, $B \approx 38.2°$, $C \approx 81.8°$ — the law of sines and cosines cosine branch.

1. SSS data: no side–angle pair exists, so the cosine half of the law of sines and cosines leads.
2. $\cos A=\dfrac{b^2+c^2-a^2}{2bc}=\dfrac{25+64-49}{80}=\dfrac{1}{2}$, so $A = 60°$.
3. $\cos B=\dfrac{a^2+c^2-b^2}{2ac}=\dfrac{49+64-25}{112}=\dfrac{11}{14}$, so $B \approx 38.2°$.
4. $C = 180° - 60° - 38.2° = 81.8°$. The law of sines and cosines angle check: $60 + 38.2 + 81.8 = 180$.

Both cosines came out positive, so every angle is acute. A negative cosine would flag the obtuse corner on sight — another law of sines and cosines tell.

Problem. Back at the canyon: $a = 30$, $b = 54$, $C = 46°$. Solve the triangle. (Granville, 1908)

Answer: $c \approx 39.6$, $A \approx 33.1°$, $B \approx 100.9°$ — the full law of sines and cosines handoff.

1. Two sides and the included angle: the law of cosines enters first. $c^2 = 30^2 + 54^2 - 2(30)(54)\cos 46° \approx 1565.3$, so $c \approx 39.6$.
2. A side–angle pair now exists ($c$ with $C$), and the sine half of the law of sines and cosines takes over. Aim at the angle opposite the smaller side: $\sin A=\dfrac{30\sin 46°}{39.6}\approx 0.545$, so $A \approx 33.1°$.
3. Subtract for the last angle: $B = 180° - 46° - 33.1° = 100.9°$. Letting $180°$ do the work dodges the trap arcsin sets for obtuse angles.
4. Check: $33.1 + 100.9 + 46 = 180$ — the law of sines and cosines cross-check passes. The 1908 tables printed 39.66, modern calculators 39.56 — $39.6$ covers both.

Two laws, one solution — the team play that gives the law of sines and cosines its name, and the canyon cabin its distance.

Problem. A lookout tower is sighted: $a = 24.31$ ft, $A = 45°18'$, $B = 22°11'$. Solve the triangle. (Wentworth & Smith, 1911)

Answer: $C = 112°31'$, $b \approx 12.9$ ft, $c \approx 31.6$ ft — the law of sines and cosines sine branch alone.

1. Two angles given, so the third is free: $C = 180° - 45°18' - 22°11' = 112°31'$.
2. Two side–angle pairs now exist, and the sine half of the law of sines and cosines does everything: $b=\dfrac{a\sin B}{\sin A}=\dfrac{24.31\sin 22°11'}{\sin 45°18'}\approx 12.9$.
3. $c=\dfrac{a\sin C}{\sin A}\approx 31.6$. On a calculator, $\sin 112°31'$ may be typed as $\sin 67°29'$, its supplement — a law of sines and cosines calculator habit.
4. Check: $b$ is the shortest side and faces the smallest angle, $22°11'$ — the order matches, a law of sines and cosines habit worth keeping.

The whole law of sines and cosines method in four lines:

1. SSS or SAS — the law of sines and cosines cosine half first; sines finish the angles.
2. AAS or ASA — the law of sines and cosines sine half alone, after $180°$ frees the third angle.
3. SSA — the law of sines and cosines wobble row: sines plus the supplement test, counting 0, 1, or 2 triangles first.
4. Any angle hunt without a pair — let cosines report it, the law of sines and cosines safety rule: a cosine never hides a supplement.

These five trip up most first attempts at the law of sines and cosines.

1. Three angles, no side. The shape is fixed but the size is not; infinitely many similar triangles fit, so no law of sines and cosines pass can finish it.
2. Forcing SAS or SSS into the sine law. Without a side opposite a known angle, every ratio in $\frac{a}{\sin A}=\frac{b}{\sin B}$ holds two unknowns — the proportion cannot start.
3. Trusting arcsin blindly. $\sin^{-1}$ only reports acute angles; in SSA the partner $180° - B$ shares the same sine and must be tested.
4. Hunting an SSS angle through sines. Two angles share one sine, so the detour breeds ambiguity; the law of sines and cosines cosine half hands every value exactly one angle.
5. Pinning a near-right angle by its sine. Near $90°$ the sine changes too slowly to fix the angle. Compute the smallest angle first and subtract, as Example 2 produced $B = 100.9°$.

Five traps, one cure: read the case before touching the calculator — the entire law of sines and cosines discipline.