Right Triangle Trigonometry (SOH-CAH-TOA)

A right triangle has one 90 degree angle, so once you fix one of the other two acute angles, the shape of the triangle is locked in — the three angles are set, and by similar-triangle reasoning every triangle with that same acute angle has the same side ratios no matter how big it is. Those fixed ratios are what sine, cosine, and tangent measure.

SOH-CAH-TOA names the three ratios relative to the acute angle you pick. Sine is Opposite over Hypotenuse, cosine is Adjacent over Hypotenuse, tangent is Opposite over Adjacent. 'Opposite' is the leg across from the angle, 'adjacent' is the leg touching the angle that isn't the hypotenuse, and the hypotenuse is always the longest side, across from the right angle. Pick the angle first, then label the other two sides relative to it — the same leg can be opposite one angle and adjacent to the other.

Once you know the ratio and one side, you can solve for another side with algebra: multiply or divide to isolate the unknown length, or use an inverse trig function if you're solving for the angle itself. This is the main tool for height-and-distance problems — angles of elevation or depression, ladders against walls, flagpoles — and it links forward to the Pythagorean theorem for finding the third side once two are known, and to circles, where the same ratios reappear defining coordinates on the unit circle.