Sine, cosine, and tangent, as most students first learn them, are defined in terms of a right triangle's sides — but most triangles encountered in surveying, navigation, and engineering aren't right triangles at all. The Law of Sines and the Law of Cosines extend trigonometry's core relationships to any triangle, letting you calculate an unknown side or angle in a general triangle from whatever measurements you already have.
The Law of Sines: a ratio that stays the same across the whole triangle
The Law of Sines states that in any triangle, the ratio of a side's length to the sine of its opposite angle is the same for all three sides — a single constant ratio holding across the entire triangle, regardless of the triangle's specific shape. This is useful whenever you know two angles and one side (letting you find the remaining sides), or two sides and a non-included angle. It's a direct generalisation of right-triangle trigonometry: apply it to a right triangle specifically, and it reduces to exactly the same relationships that basic sine and cosine definitions already describe, just expressed in a form that keeps working once the right angle is gone.
The Law of Cosines: Pythagoras with a correction term
The Law of Cosines generalises the Pythagorean theorem itself, which only applies to right triangles. It states that the square of one side equals the sum of the squares of the other two sides, minus a correction term involving twice their product and the cosine of the angle between them. When that angle happens to be exactly 90 degrees, cosine of 90 degrees is zero, the correction term vanishes entirely, and the formula collapses precisely back into the familiar Pythagorean theorem — making the Law of Cosines a strict generalisation, with ordinary Pythagoras as a special case rather than a separate, unrelated rule. It's particularly useful when you know all three sides and want an angle, or two sides and the angle between them, cases the Law of Sines alone can't directly resolve.
What we're still unsure about
Both laws are fully proven, classical trigonometric results with no live mathematical dispute attached to them. The genuinely practical complication, in applied fields like surveying and navigation, isn't the mathematics itself but measurement — real-world angle and distance measurements always carry some margin of error, and that error propagates through these formulas in ways that aren't always intuitive, meaning small measurement inaccuracies can sometimes produce disproportionately large errors in a calculated result, particularly for certain triangle shapes. Managing and minimising that error propagation, rather than the underlying trigonometric identities, is where much of the genuine skill in applied surveying and navigation actually lies.
This sits inside Law of Sines & Law of Cosines, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.