Ratios That Never Move

Where it usually goes wrong

SOHCAHTOA is memorized per-triangle, as three divisions to redo on every new drawing, so nobody notices that the SAME angle always returns the same three numbers no matter how the triangle happens to be drawn — sin, cos and tan read as facts about a picture's size rather than facts about its shape, which is why a bigger, more carefully drawn triangle feels like it ought to need re-measuring.

Try this

• Drag the radius slider back and forth without touching the angle. The triangle visibly grows and shrinks, but sin θ, cos θ and tan θ in the readout do not change by even a hundredth.
• Press "45°", then drag the radius slider to its largest value. The triangle is now much bigger than it started, and sin and cos are still both exactly 0.71 — a ratio, not a length.
• Drag the point freely until it lands on the dashed unit circle (radius reads 1.00). At that exact spot, the point's own (x, y) coordinates equal (cos θ, sin θ) — no division left to do at all.

Where this goes next

This is why a calculator's sin button takes only an angle and no side lengths: it is returning the coordinate a point at that angle makes on a circle of radius 1, the same number this picture shows directly the moment the radius slider reaches 1.