Sine and Cosine of Complementary Angles

A right triangle has one 90 degree angle and two acute angles that add up to 90 degrees, so they are complementary. Pick one of the acute angles and label the sides: the side across from it is opposite, the side next to it (not the hypotenuse) is adjacent. Now look at the triangle from the other acute angle's point of view. The side that was opposite the first angle is now adjacent to the second one, and the side that was adjacent is now opposite. The hypotenuse never changes. Since sine is opposite over hypotenuse and cosine is adjacent over hypotenuse, the sine of one angle and the cosine of the other end up being the exact same ratio of the exact same two sides.

That gives the identity sin(x) = cos(90 - x) for any acute angle x. It works in both directions: cosine of an angle also equals sine of its complement. This is not a coincidence you have to re-derive each time, it is just a restatement of 'the two acute angles share sides, swapped.'

This identity shows up fused with linear equations: you get two angle expressions set equal through sine and cosine, and solving means setting those expressions to add to 90, then solving for the variable. It sits right next to plain right-triangle trig and the Pythagorean theorem, since all of it comes from the same picture of one triangle, two acute corners, three sides.