Synchronized instrument

Visual Trigonometry

Angle unit

Exact geometry

Standard angles

Current construction π/4 · 45° · exact

When enabled, dragging near a standard-angle guide snaps to it.

Live equivalence

One point, four readings

Polar (r, θ) = (1, π/4)
Cartesian (x, y) = (0.707, 0.707)
Complex z = 0.707 + 0.707i
Exponential z = reiθ
θ
π/4
cos θ
0.707
sin θ
0.707
turns
0

Identity lenses

Selected lens

Coordinates / Euler

foundation
Algebra
z = reiθ = x + iy
Exact geometry 45° isosceles-right triangle Quadrant I · signs (+, +) a² + a² = 1 → a = √2/2 cos θ = √2/2 · sin θ = √2/2 ≈ (0.707, 0.707)

Equal legs in a right triangle with unit hypotenuse have length square root of two over two.

Geometry

The projections are the real and imaginary coordinates.

Complex form
eiθ = cos θ + i sin θ
Trigonometric result
x = r cos θ, y = r sin θ

The geometry, complex form, and trigonometric result use the same shared point.

Synchronized circle and waveform views

Direction becomes position

Circle / complex plane

  • u
  • z

Drag the point u around the circle

At theta pi over four, the unit point and radius-one polar point coincide at approximately x 0.707 and y 0.707.

Horizontal coordinate

Cosine

Drag the wave cursor

The cosine plot marks the current point at theta pi over four with value approximately 0.707.

Vertical coordinate

Sine

Drag the wave cursor

The sine plot marks the current point at theta pi over four with value approximately 0.707.