Out of the Right-Angle Cage
From the first guide in this rung you can take any right triangle, name a sharp angle, and read off its sides with SOHCAHTOA. That is genuinely powerful — but look closely and you will see the whole machine leans on one ingredient: the right angle. The hypotenuse, the opposite, the adjacent — those labels only make sense when one corner is exactly 90 degrees. Tilt that corner away from square and the names lose their meaning, and solving the triangle the old way stalls.
Yet most triangles you meet — a surveyor's plot, a roof truss, the triangle between two ships and a lighthouse — have no right angle anywhere. We need tools that solve a triangle from whatever scraps we happen to know: maybe two angles and a side, maybe two sides and the angle between them, maybe all three sides. Two laws cover every one of those cases. They are the natural sequel to SOHCAHTOA, and the secret is that we will not abandon right triangles at all — we will keep slicing the general triangle into right ones with a single dropped height.
The Law of Sines: One Ratio That Every Side Obeys
Drop a perpendicular height h from vertex C straight down onto side c. That single line cuts the triangle into two right triangles, and now the old machinery applies inside each half. In the left half, sin A = h / b, so h = b sin A. In the right half, sin B = h / a, so h = a sin B. Both expressions equal the same height, so b sin A = a sin B, which rearranges to a / sin A = b / sin B. Drop a height to a different side and the third ratio joins in — that is the whole proof.
Written out, the law of sines says a / sin A = b / sin B = c / sin C — one ratio shared by all three side-and-angle pairs. Try a quick AAS solve: say A = 40 degrees, B = 60 degrees, and side a = 8. The third angle is C = 180 - 40 - 60 = 80 degrees. Then b = 8 times sin 60 / sin 40 = 8 times 0.866 / 0.643, which is about 10.78, and c = 8 times sin 80 / sin 40, about 12.25. Three angles and one side gave us the whole triangle in two short lines.
Read the law of sines as a statement of perfect proportion: each side is to the sine of its opposite angle in one shared ratio, the same for all three. It is the right tool the moment you can pair a side with the angle across from it — the AAS and ASA cases, where you know two angles (so the third is free, since the angles sum to 180 degrees) and any one side. That common ratio is not just bookkeeping, either: it equals the diameter of the triangle's circumscribed circle, a quiet bridge back to circle geometry.
The Ambiguous Case: When the Data Allows Two Triangles
The law of sines has one honest trap, and skipping it is how careless students lose marks. Suppose you know two sides and an angle that is not between them — say sides a and b and the angle A opposite a. You swing side a like a hinged door from the end of side b, trying to reach the base line. Depending on how long a is, the door can strike the base line in two places, in exactly one place, or not reach it at all. So the same numbers can describe two different triangles, one triangle, or none — this is the ambiguous case, the SSA situation.
The arithmetic of why is simple and worth seeing. When you solve sin B = b sin A / a, the inverse-sine button on the calculator only ever hands back the acute angle — but its supplement (180 degrees minus that value) has the very same sine, from the unit circle symmetry you saw two guides back. If that supplementary angle still leaves the angle sum under 180 degrees, it gives a second, genuinely valid triangle. This is not a flaw in the law; it is the law faithfully reporting that SSA simply does not pin a triangle down, unlike SAS or SSS which always do.
The Law of Cosines: Pythagoras With a Correction
The law of sines needs a side paired with its opposite angle, so it stumbles in two common situations: when you know two sides and the angle squeezed between them (SAS), or when you know all three sides and no angle at all (SSS). For those, reach for the law of cosines. It looks heavier, but it is really just the Pythagorean theorem with one extra term bolted on to account for the angle not being 90 degrees.
Law of cosines: c^2 = a^2 + b^2 - 2 a b cos C
When C = 90 deg, cos C = 0, and it collapses to:
c^2 = a^2 + b^2 (the Pythagorean theorem)
Example (SAS): a = 5, b = 7, C = 60 deg
c^2 = 25 + 49 - 2*5*7*cos 60
= 74 - 70 * 0.5 = 74 - 35 = 39
c = sqrt(39) = 6.24Look at the correction term -2 a b cos C and the honesty of it pours out. If C is a right angle, cos C = 0 and the term vanishes, leaving plain old a^2 + b^2 = c^2. If C is acute, cos C is positive so we subtract something, making c shorter — the side across from a squeezed-in angle is short, exactly as your eye expects. If C is obtuse, cos C is negative, the minus-times-minus adds on instead, and c grows long. The single formula quietly knows the difference between acute and obtuse triangles, with no separate cases to memorise.
And the law of cosines runs in reverse. Rearrange it to cos C = (a^2 + b^2 - c^2) / (2 a b), feed in three known sides, and out comes an angle. There is no ambiguity here the way there was with sines: cosine is negative for obtuse angles and positive for acute ones across the whole range 0 to 180 degrees, so each cosine value names exactly one triangle angle. That is why the law of cosines is the trustworthy tool for the SSS case, where three sides must determine every angle uniquely — the very ambiguity that plagued SSA simply cannot arise.
Choosing Your Tool, and a Bonus for Area
With both laws in hand, solving any triangle becomes a question of matching the tool to what you were given. The deciding question is always the same: can you pair a side directly with the angle opposite it? If yes, the law of sines is faster. If not — if your known angle is wedged between two known sides, or you have only sides — the law of cosines is the way in, often used once to find a third side and then handing off to the law of sines for the remaining angles.
- Inventory what you know. Count the given sides and angles, and note which angle sits opposite which side.
- Two angles and any side (AAS / ASA): find the third angle from the 180-degree sum, then use the law of sines for the missing sides.
- Two sides and the included angle (SAS): use the law of cosines once to get the opposite third side, then finish with the law of sines or cosines.
- All three sides (SSS): use the law of cosines in its rearranged form to read off each angle, with no ambiguity to worry about.
- Two sides and a non-included angle (SSA): use the law of sines, then deliberately test the supplementary angle for a possible second triangle.
One graceful bonus falls out of all this. Once you know two sides and their included angle, you can get the triangle's area without ever finding a height: the sine area formula, Area = (1/2) a b sin C. It comes straight from base-times-height-over-two once you notice the height equals b sin C — the same dropped perpendicular that gave us the law of sines in the first place. Sides, angles, and area all flow from that one honest little line you drew from a vertex. SOHCAHTOA opened the door; these laws walk you through every room beyond it.