Phase Shift Explorer — sin, cos & Unit Circle

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sin(x + φ)

PHASE SHIFT φ 0°

▲

▼

any°
±45

−360°+360°

AMPLITUDE A 1.00

0.13.0

FREQUENCY ω 1.00

0.1×T=2π5×

SIN · COS · UNIT CIRCLE

PHASE SHIFT EXPLORER · φ ∈ (−∞, +∞)

sin(x + φ)

cos(x + φ)

Unit Circle

Arc Length

Small θ

+1 +0.5 0 −0.5 −1

0°

φ shift

drag to rotate

PHASE ANGLE φ — any value

▲

▼

type any°
↑↓ = ±1

← drag left/right · scroll · unlimited →

PRESETS

SPEED

FUNCTION

VALUE AT x=0

sin(φ)

PEAK AT x=

max = +1

EQUIVALENT

FULL CYCLES

× 360°

sin(x)

cos(x) ref

sin(x+φ)

+1 +0.5 0 −0.5 −1

0°

φ shift

drag to rotate

PHASE ANGLE φ — any value

▲

▼

type any°
↑↓ = ±1

← drag left/right · scroll · unlimited →

PRESETS

SPEED

FUNCTION

VALUE AT x=0

cos(φ)

PEAK AT x=

max = +1

EQUIVALENT

FULL CYCLES

× 360°

cos(x)

sin(x) ref

cos(x+φ)

+1 +0.5 0 −0.5 −1

−1 −0.5 0 +0.5 +1 ← cos(θ) = x →

↑ sin(θ) = y ↑

ANGLE θ (degrees — any value)

▲

▼

type any°
↑↓ = ±1

← drag left/right · scroll · unlimited →

NOTABLE ANGLES

SPEED

▸ sin(θ) = y-coordinate (vertical)

+1 0 −1

▸ cos(θ) = x-coordinate (horizontal)

+1 0 −1

cos(θ) = x

1.000

horizontal projection
onto x-axis

sin(θ) = y

0.000

vertical projection
onto y-axis

θ (normalised)

0°

within 0°–360°

full rotations

× 360°

IDENTITY AT THIS ANGLE

sin(0°)=0 · cos(0°)=1

▸ PHASE SHIFT INSIGHT

CHOOSE RADIUS r — YOUR CIRCLE (unit circle stays fixed at r=1)

CHOOSE ANGLE θ — drag the point on YOUR CIRCLE or use the wheel

60°

Arc length s = r·|θ| is always positive — drag CW to visualise clockwise rotation.

RADIUS r

1.000

ANGLE θ

60°

1.047 rad

↺ CCW

ARC s = r·|θ|

1.047

always ≥ 0

UNIT ARC s₁=θ

1.047

🔒 r=1 fixed → s=θ

KEY INSIGHT — s = r · θ

Both circles share the same angle θ. The unit circle arc s₁ = θ (numerically equal). Your circle arc s = r × s₁ — it scales by r.

ANGLE θ — drag slider

30.0°

SMALL ANGLE APPROXIMATIONS

sin θ ≈ θ

actual sin θ

—

approx θ

—

cos θ ≈ 1

actual cos θ

—

approx 1

1.00000

—

s = r·θ

arc s

—

chord

—

SPEED

🔍 ZOOM 1×

① sin θ ≈ θ

The opposite side of the right triangle = r·sin θ.
When θ is tiny, the arc and the opposite side are nearly equal.
Since arc = r·θ, we get sin θ ≈ θ.
Taylor: sin θ = θ − θ³/6 + … ≈ θ

② cos θ ≈ 1

The adjacent side of the triangle = r·cos θ.
When θ → 0, the radius barely tilts — the horizontal projection stays ≈ r.
So cos θ ≈ 1 (next term: 1 − θ²/2).
Taylor: cos θ = 1 − θ²/2 + … ≈ 1

③ s = r · θ

Arc fraction of full circle:
s = 2πr · (θ/2π) = r · θ
When θ→0, chord ≈ arc ≈ r·θ,
so all three sides converge.
chord = 2r·sin(θ/2) ≈ rθ

ERROR TABLE (r=1)

θ	sinθ err	cosθ err	arc err
1°	<0.01%	0.015%	<0.001%
5°	0.13%	0.38%	0.13%
10°	0.51%	1.52%	0.51%
20°	2.06%	6.03%	2.06%
30°	4.72%	13.4%	4.72%