Opposite and adjacent seem like names of the sides. They are not: they are names of the relationship between a side and the angle you chose to look at. Change the angle and the two names change sides, without the triangle moving a millimeter. The hypotenuse is the only one that doesn't move — it is always the side opposite the right angle.
Do: change the shape and see the three sides change together. Press view from the other angle and see the names switch places. Then adjust the size: all lengths change, and no ratio changes — that’s exactly why the ratios have a name.
| length | the ratio it forms | |
|---|---|---|
| opposite leg | ||
| adjacent leg | ||
| hypotenuse | is the ruler for the other two | |
| tangent | — |
1. Change angle. The triangle doesn’t change — no side moves. But what was opposite becomes adjacent and vice-versa, because the names describe the position in relation to the angle viewed. Those who memorized SOH-CA-TOA without seeing this treat the names as if they were of the side, and err when the drawing is rotated.
2. And then sine becomes cosine. The two acute angles add up to a right angle —
they are complementary. Viewing from the other, the opposite and the adjacent swap,
so the ratio that was the sine becomes the cosine:
sin θ = cos(90° − θ). That’s where the "co" in cosine comes from: it is
the sine of the coplement. Turn on show complement and see the two angles
closing the right angle.
3. Change the size, and look at the two columns. The lengths all change. The ratios do not change at all. That's why sine and cosine are properties of the angle, and not of the triangle — and that's why the school formula works for any triangle, and not just the one in the book.
4. The hypotenuse is the only one that doesn't change its name. It is defined by the right angle, not by the angle you are looking at — it is always the side opposite to it, and always the largest of the three.
Set the size to 1. Now the hypotenuse is one, and the two ratios become the
actual lengths: the opposite is the sine, the adjacent is the cosine. This is
exactly the triangle drawn inside the o-circulo.html, with the hypotenuse in the place of the radius — and the colors here are the same on purpose.
It is also the triangle in the taxa.html: there the adjacent is 1 and the opposite
is f′, and so f′ = opposite/adjacent = tan θ.
Local, self-contained instrument: no network, no external library. Slice of the same
family as o-angulo, o-circulo, par-vira-ponto and
taxa.