Sine and Cosine of Complementary Angles

A right triangle has one 90 degree angle and two acute angles that add up to 90 degrees. Pick one acute angle, call it A. The other acute angle is then 90 minus A. Sine of A is opposite over hypotenuse. Cosine of 90 minus A is adjacent over hypotenuse, but adjacent to the second angle is the same side that was opposite the first angle. Same ratio, same value. That is the whole reason sin 20 degrees equals cos 70 degrees exactly: 20 and 70 are complementary.

The mechanism is just relabeling. Stand at angle A and the leg across from you is opposite; stand at the other acute angle and that same leg is now the one next to you, adjacent. The hypotenuse never changes. So every sine becomes a cosine and every cosine becomes a sine the moment you switch which acute angle you are measuring from. This is why you can solve equations like sin(2x + 6) = cos(x + 9) by setting the two angle expressions to add to 90, instead of ever touching an actual triangle or a calculator.

This complementary relationship sits right next to the Pythagorean identity, sin squared A plus cos squared A equals 1, and the two are easy to blur together. They are not the same fact: one links two different angles, the other links two ratios of the same angle. Keeping that boundary clear also sets up the two special right triangles and the general trig ratios, since cofunction swaps show up again whenever a problem hands you one ratio and expects the complementary one.