Three ratios
Drag the orange corner, or use the sliders. The angle slider changes the shape of the right triangle. The size slider makes the same shape bigger or smaller. Watch the three side lengths change, then watch the three ratios underneath. One of the sliders does nothing to them.
Mission. Set the angle to 30° and read off sin 30°. Now change the size. What happens to it? Then find the angle where tan is exactly 1, and say why it has to be that angle.
Answer
sin 30° = 0.5 at every size: the opposite side is always half the hypotenuse. tan = 1 at 45°, because tan is opposite ÷ adjacent, and those two sides are equal only when the triangle is half a square.
opposite (across from the angle) adjacent (next to the angle) hypotenuse (the longest side)
Side lengths are rounded to 2 decimal places, so a calculator may disagree with the last digit of a ratio.
What's going on
Two right triangles with the same angle are the same shape. One is a scaled copy of the other. Scaling multiplies every side by the same number, so when you divide one side by another, that number cancels. The ratio depends on the angle and on nothing else.
That makes each ratio a way to measure the angle, and each has a name. Sine is opposite ÷ hypotenuse. Cosine is adjacent ÷ hypotenuse. Tangent is opposite ÷ adjacent. The dashed line shows it: slide the size and the corner runs along the same line, because every triangle with this angle fits under it.
Tangent is rise over run, so it is the slope of that dashed line. A ramp that rises 1 for every 12 along has tan = 1 ÷ 12 ≈ 0.083. Slide the angle down until tan is close to that. It is about 5°, which is gentler than most people guess.
Sheets: Three ratios that measure an angle and How steep is too steep?