Back to Geometry
Geometry › Trigonometry
Trigonometry
The math of triangles — three side ratios that unlock angles, distances, waves, music and orbits.
Unit circle · sin, cos, tan
Spin the radius. The dot's coordinates ARE (cos θ, sin θ).
θ = 45°
45°
sin θ
0.707
cos θ
0.707
tan θ
1.000
SOH · CAH · TOA
sin = O/H · cos = A/H · tan = O/A
Right-triangle trig
For a right triangle with one acute angle θ:
- Opposite — the side across from θ.
- Adjacent — the side next to θ (not the hypotenuse).
- Hypotenuse — the longest side, opposite the right angle.
SOH · CAH · TOA
Sin = Opposite / Hypotenuse · Cos = Adjacent / Hypotenuse · Tan = Opposite / Adjacent
The unit circle
Place a point on a circle of radius 1 at angle θ from the positive x-axis. Its coordinates ARE (cos θ, sin θ). Sin and cos extend smoothly past 90° this way — and become the basis for waves, oscillations and rotations.
Beyond right triangles
- Law of sines:
a/sin A = b/sin B = c/sin C - Law of cosines:
c² = a² + b² − 2ab·cos C(Pythagoras + a correction)
Where it shows up
Sin and cos describe waves — sound, light, AC electricity, the swing of a pendulum, the rise and fall of tides. Trig is the bridge between rotation and oscillation.