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Trigonometry · 01 · Honest ratios · 8 min

Sine, cosine & tangent

Fix an angle in a right triangle and something remarkable happens: the ratios between the sides lock into place, no matter how big the triangle is. Those locked ratios have names — sine, cosine, tangent — and they're the whole of trigonometry.

Build the intuition

Same angle, same ratios — guaranteed

Draw a 30° right triangle the size of a stamp or the size of a field: the side opposite the angle is always exactly half the hypotenuse. Angles control proportions, not sizes. That's why one small table of ratios (or one calculator button) serves every triangle ever drawn.

The three names

For an angle θ: sine is opposite ÷ hypotenuse, cosine is adjacent ÷ hypotenuse, tangent is opposite ÷ adjacent. SOH-CAH-TOA compresses it to a breath. Each ratio answers a different question: how tall (sin), how far along (cos), how steep (tan).

sin⁡θ=opphypcos⁡θ=adjhyptan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}} \quad \cos\theta = \frac{\text{adj}}{\text{hyp}} \quad \tan\theta = \frac{\text{opp}}{\text{adj}}

The trade: one angle + one side = everything

Know a single angle and a single side of a right triangle, and the ratios hand you every other measurement. That's the engine inside every “how tall is the tower if I stand here?” problem — and inside the instruments that answer it professionally.

See it move

InteractiveThe circle that makes waves
0.9
At angle 0.9 rad: height sin θ = 0.78, horizontal cos θ = 0.62. Walk the full circle and the height traces one complete sine wave — triangles and waves are the same picture.

The triangle lives inside a circle of radius 1: the orange height is sine, the teal base is cosine. Turn the angle and watch the ratios breathe.

A worked example

Height of a tree, no climbing

  1. From 20 m away, you sight the treetop at a 35° angle.

  2. Steepness is tangent's department:

    tan⁡35°=h20\tan 35° = \frac{h}{20}

  3. Solve:

    h=20tan⁡35°≈20(0.700)=14 mh = 20 \tan 35° \approx 20 (0.700) = 14 \text{ m}

  4. Add your eye height (~1.6 m): the tree stands about 15.6 m. One angle traded for one height.

Out in the world

Aircraft descent angles

A standard approach glide is about 3°. Air traffic systems and pilots constantly convert between distance-to-runway and required altitude using exactly tan(3°) — trigonometry keeping descents smooth and safe on every flight you've taken.

Common confusion, cleared

“sin is a button that does something mysterious to a number.”

sin θ is just a ratio of two triangle sides — a number like 0.5. The button is a lookup of geometry, not an operation on digits.

“Opposite and adjacent are fixed sides.”

They're relative to the angle you're using: switch to the other acute angle and they swap roles. Always orient yourself from θ first.

Recap

  • An angle locks the side ratios of a right triangle — at every size.
  • sin = opp/hyp, cos = adj/hyp, tan = opp/adj (SOH-CAH-TOA).
  • One angle plus one side unlocks the entire triangle.