SpecEnvoy

EN·中文

Sin, Cos & Tan

Three words that scare a lot of people — but each one is just a ratio of two sides in a right-angled triangle. Let's make them move.

The right-triangle idea: SOH-CAH-TOA

Pick one of the non-right angles in a right-angled triangle and call it θ ("theta"). The three sides get names relative to θ:

• Hypotenuse — the longest side, opposite the right angle.
• Opposite — the side across from θ.
• Adjacent — the side next to θ (that isn't the hypotenuse).

Say it out loud a few times — "soh-cah-toa" — and you'll never lose which is which.

Now see it on the unit circle

Here's the trick that ties everything together. Draw a circle of radius 1 (a "unit circle"). Put a point on it and spin it around by angle θ. Then the point's coordinates are exactly cos θ, sin θ — cosine is how far across, sine is how far up.

Things to notice as you drag:

• The green height is sin θ; the blue width is cos θ. The slanted radius is the hypotenuse — and it's always length 1.
• Both stay between −1 and 1. That's why sin and cos can never be bigger than 1.
• Past 90° the point moves into other quadrants, so sin or cos go negative — the circle keeps the definitions working for any angle, not just triangles.
• Turn on "Show tan line": tan θ = sin θcos θ, and you can see it shoot off to infinity as the angle nears 90° (because cos θ → 0).

Some angles worth memorising

Tap the snap buttons in the demo (30°, 45°, 60°…) to land exactly on these and check.

Where sine waves come from

Keep spinning the point steadily and just watch its height over time. That height — the sine — rises, falls, goes negative, and repeats. Plot it and you get the famous sine wave. Press play:

The dashed line connects the spinning point to the wave — they're always at the same height, because they're the same number: sin θ. This is why trig describes anything that repeats: sound, light, springs, tides, even your heartbeat on a monitor. Tick "Show cosine wave" to see cos doing the same thing, just a quarter-turn ahead.