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Virtual Math LabGeometry, Graphing & Trigonometry

Mathematics Visualization Lab

Manipulate geometric constructions, function parameters, and unit circle angles with real-time feedback.

Euler Line System
Tool: SELECTΔABC Area: 13.97 u² (Acute)
-6-4-2246-4-224O (Circumcenter)A(-3.0, -1.5)B(3.5, -1.5)C(0.5, 2.8)

Geometric Measurements & Analysis

Primary Triangle ΔABC
AcuteScalene
Side a (BC)5.24
Side b (AC)5.54
Side c (AB)6.50
∠A (at A)50.9°
∠B (at B)55.1°
∠C (at C)74.0°
Perimeter (a + b + c)17.29 units
Area (Heron's Formula)13.97 units²
Centers Coordinates
Centroid G: (0.33, -0.07)
Circumcenter O: (0.25, -0.57), R = 3.38
Incenter I: (0.40, 0.12), r = 1.62
Orthocenter H: (0.50, 0.94)
Construction History
1.Initialized Triangle ABC
12:00:00
Curvature / Opening (a)1.0
Linear Coefficient (b)0.0
Constant (c = y-intercept)0.0
Tracer: x = 2.00, f(x) = 4.00
-6-4-2246-6-4-2246Vertex (0.0, 0.0)
Current Mathematical Model
y = 1.0x² + 0.0x + 0.0
Y-Intercept(0, 0.00)
Roots (x-intercepts)0.00
Parabola Vertex(0.00, 0.00)
Geometric & Algebraic Meaning

Parabola opens upward with minimum vertex at (0.00, 0.00). Axis of symmetry: x = 0.00. One real repeated root at x = 0.00.

Angle θ45° (π/4)
Quick Standard Angles
Speed:
ASTC Quadrant Sign Rules
All-Silver-Tea-Cups
Quadrant I (0° – 90°)Active
sin +
cos +
tan +
Quadrant II (90° – 180°)
sin +
cos -
tan -
Quadrant III (180° – 270°)
sin -
cos -
tan +
Quadrant IV (270° – 360°)
sin -
cos +
tan -
Active Angle Insight:

At θ = 45°, the horizontal coordinate is x = +0.71 and the vertical coordinate is y = +0.71. A: All trig functions are positive (x > 0, y > 0).

θ = 45° (π/4)cos θ = √2/2sin θ = √2/2tan θ = 1
IIIIIIIVP(cos θ, sin θ)
Connected Wave Projection: Sine Curve y = sin(θ)
+1.0-1.00π/2 (90°)π (180°)3π/2 (270°)2π (360°)(π/4, 0.71)
The Connected Nature of Mathematics

From Constructions to Functions & Waves

Mathematics links geometric shapes, algebraic equations, and circular periodic motion. In this lab, observe how geometric right triangles give rise to the unit circle coordinates (cos θ, sin θ), which unroll continuously to generate trigonometric wave functions.