The trig ratios you learned in middle school (sine, cosine, tangent) only worked under one condition: a right triangle. But most triangles in the real world don't have a right angle. Finding the distance to a mountain peak, the distance between two ships, or the area of a plot of land by triangulation — these problems almost always involve triangles that aren't right triangles. That's where the law of sines and the law of cosines come in. With just these two laws, no matter what shape a triangle is, knowing only a few of its angles and sides lets you calculate everything else.
Let's start with the law of sines. If a triangle's three angles are A, B, C, and their respective opposite sides are a, b, c, the following relationship always holds.
The heart of the law of sines is that this ratio is always the same. This value even matches exactly the diameter (2R) of the circle circumscribed around the triangle. It's an expression that shows a natural relationship: the larger an angle, the longer its opposite side, and the larger the triangle overall, the larger its circumscribed circle. The law of sines is especially useful when you know two angles and the side opposite one of them (an angle-angle-side situation, like ASA or AAS). Once you know two angles, the third is found instantly by subtracting from 180°, and with just one opposite side you can set up a proportion to find both remaining sides.
But what if you only know two sides and the angle between them (an SAS situation)? In that case, you can't use the law of sines directly, because you don't yet have a single complete side-angle pair. That's where the law of cosines comes in.
The law of cosines is: c² = a² + b² − 2ab·cosC (where C is the angle between sides a and b). Look closely and this should look familiar. If C is 90°, then cosC = 0, and the formula becomes c² = a² + b² — that's exactly the Pythagorean theorem! In other words, the law of cosines is the Pythagorean theorem extended to any angle, not just right angles. If C is less than 90°, the term −2ab·cosC becomes negative, making c shorter; if C is greater than 90°, this term becomes positive, making c longer — matching the intuition that a sharper angle means a shorter opposite side.
To sum up: if you know two or more angles and one opposite side, use the law of sines; if you know two sides and the angle between them (or all three sides), use the law of cosines. Both laws completely free you from the middle-school limitation of trig ratios, which only worked "if there's a right angle." On the activity page, try changing angles and side lengths yourself with sliders, and you'll see for yourself that fixing just a few of the three values locks the whole triangle into a single shape.