01
Right triangles
How does an angle determine side ratios?
Sine, cosine, and tangent begin as stable ratios among corresponding sides of similar right triangles.
Angle · ratio · rotation · periodicity
Trigonometry
Trigonometry turns angles into ratios, ratios into coordinates, and coordinates into periodic functions. The same structure connects right triangles, the unit circle, rotation, oscillation, and wave-like behavior.
Concept pathway
One idea keeps changing representation.
Start with a triangle ratio, move it onto a circle, then let the angle advance continuously. Trigonometry becomes much less mnemonic once those three views are connected.
02
Unit circle
How do those ratios work through a full rotation?
The point at angle θ has coordinates (cos θ, sin θ), extending trigonometric functions beyond acute triangle angles.
03
Periodic signals
What happens when rotation unfolds through time?
Circular coordinates oscillate. That periodic structure makes sine and cosine natural models for waves, cycles, and rotation.
Right-triangle solver
ratios scale with the triangleAngle θ38°
Hypotenuse12.0 units
sin θ
0.616opp / hypcos θ
0.788adj / hyptan θ
0.781opp / adjChanging the hypotenuse scales every side but leaves the three ratios unchanged. Changing the angle changes the ratios. That separation is the core reason trigonometric functions can describe shape independent of size.
Unit-circle coordinates
radius = 1Angle
45°
0.25π rad
cos θ · x0.707
sin θ · y0.707
tan θ = sin θ / cos θ1.000
The rotating point has coordinates (cos θ, sin θ). Triangle ratios become coordinates on a circle, which makes the same functions work for every rotation instead of only acute right triangles.
Circle → signal
A sine wave is circular motion read one coordinate at a time.
The ambient construction behind this page continuously projects the vertical coordinate of a rotating point into a sine wave. Cosine is the horizontal projection of the same motion.
01Coordinate
(cos θ, sin θ)The unit-circle point at angle θ.
02Pythagorean
sin²θ + cos²θ = 1The unit circle is x² + y² = 1 with x = cos θ and y = sin θ.
03Tangent
tan θ = sin θ / cos θSlope-like ratio; undefined where cos θ = 0.
04Period
sin(θ + 2π) = sin θOne full turn returns to the same coordinate.