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Trigonometry Simulations

A collection of interactive 3D visualizations and simulations to help you master concepts in trigonometry.

Angles and their Measure - Examples & Applications

Comprehensive guide to Degrees, Radians, Coterminal Angles, Arc Length, Sector Area, and Circular Segments with interactive visualizations.

Angle Visualizer

Angle ($\theta$)135°
-720°720°
Show Reference Angle
Show Coterminal (360°)
Standard Position:135°
Reference Angle ($\alpha$):45°
Quadrant:II
0°90°180°270°

Trigonometric Functions - Examples & Applications

Comprehensive study of Sine, Cosine, Tangent, Unit Circle properties, Domain, Range, and Graph transformations.

ASTC Rule Visualizer

Hover or tap on a quadrant to see which functions are positive.

SQuadrant II
AQuadrant I
TQuadrant III
CQuadrant IV
Select a quadrant to view its properties.

Trigonometric Identities - Examples & Applications

Fundamental identities, Pythagorean identities, Sum/Difference, Double/Half Angle formulas, Co-function identities, Power-Reducing formulas, and proofs.

Pythagorean Identity Verifier

Adjust the angle to see that $\sin^2\theta + \cos^2\theta$ always equals 1.

cos(θ)sin(θ)
cos(45°)0.7071
sin(45°)0.7071
Pythagorean Theorem
cos²θ + sin²θ =1
(0.5000) + (0.5000) ≈ 1

Inverse Trigonometric Functions - Examples & Applications

Deep dive into arcsin, arccos, arctan, domain restrictions, principal values, and composition.

Principal Values

Inverse Sine (arcsin x)

Output Range[ -π/2, π/2 ]
Input Domain[-1, 1]
Valid QuadrantsI and IV
The inverse function returns the unique angle in the highlighted range that corresponds to the given ratio.
0π/2π-π/2

Trigonometric Equations - Examples & Applications

Methods for solving linear, quadratic, and multiple-angle trigonometric equations.

π2π-π-2π0.17π0.83π-1.83π-1.17π

Showing solutions for sin(x) = 0.50 in the interval [-2π, 2π].

Trigonometric equations often have multiple solutions due to their periodic nature.

Applications of Trigonometry - Examples & Applications

Solving real-world problems using SOH CAH TOA, Law of Sines, Law of Cosines, and the Law of Tangents.

Right Triangle Solver

Base (b)4
Height (a)3
Hypotenuse (c):5.00
Angle A:36.9°
Angle B:53.1°
Area:6.00
abc

Spherical Trigonometry - Examples & Applications

Principles of spherical triangles, Polar triangles, Napier's Rules, Napier's Analogies, spherical laws, and terrestrial navigation applications.

Spherical Triangle Geometry

30°
45°
20°
30°

Spherical Triangle N-P-O:
- Side NP = Colatitude (90° - φ)
- Side NO = 90°
- Angle at N = Longitude (λ)

Complex Numbers and Polar Coordinates - Examples & Applications

Introduction to polar coordinates, expressing complex numbers in polar form, multiplication, division, and De Moivre's Theorem.

Adjust Complex Number zz

Original Number (zz):

z=1.0+0.0iz = 1.0 + 0.0i
∣z∣=r=|z| = r = 1.00θ=\theta = 0°

Roots (wkw_k):

w0=1.03(cos⁡(0∘)+isin⁡(0∘))w_0 = \sqrt[3]{1.0} \left(\cos(0^\circ) + i\sin(0^\circ)\right)
Roots are spaced by 120.0°