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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 - Theory & Concepts - Trigonometry Angle Visualizer

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

Angles and their Measure - Theory & Concepts - Trigonometry Radians Degrees Converter

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

Radians & Degrees Converter

Visualize angle measurement conversion on a 2D coordinate system.

Angle (Degrees)45°
0°90°180°270°360°

Conversion Formulas

Degrees to Radians:
rad=θ∘×π180∘\text{rad} = \theta^\circ \times \frac{\pi}{180^\circ}
Calculation:45∘×π180∘=π4≈0.7854 rad45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4} \approx 0.7854\text{ rad}
Radians to Degrees:
deg=rad×180∘π\text{deg} = \text{rad} \times \frac{180^\circ}{\pi}
Calculation:0.7854×180∘π=45∘0.7854 \times \frac{180^\circ}{\pi} = 45^\circ
Angles measured counter-clockwise from the positive x-axis are positive. One full rotation corresponds to 360∘360^\circ or 2π2\pi radians.
2D Coordinate System Angle VisualizerX AxisY AxisUnit Circle HelperX (0°/2π)Y (90°/π/2)-X (180°/π)-Y (270°/3π/2)Swept Angle ArcTerminal SideOrigin (0,0)Terminal Coordinate Point(0.71, 0.71)
Angle Measure
45∘=π4 rad45^\circ = \frac{\pi}{4}\text{ rad}

Angles and their Measure - Theory & Concepts - Trigonometry Arc Length Sector Area

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

Arc Length & Sector Area Visualizer

Adjust the radius and angle to see how they govern the arc length ($s = r\theta$) and the sector area ($A = \frac12r^2\theta$).

Radius (rr)3.0 units
1.02.03.04.05.0
Angle (θ\theta)120° / 2.094 rad
0°90°180°270°360°

Step-by-Step Calculations

1. Convert Angle to Radians:
θ=120∘×π180∘=2.0944 rad\theta = 120^\circ \times \frac{\pi}{180^\circ} = 2.0944 \text{ rad}
2. Arc Length (s=rθs = r\theta):
s=3.0×2.0944s = 3.0 \times 2.0944
s=6.2832 unitss = 6.2832 \text{ units}
3. Sector Area (A=12r2θA = \frac{1}{2}r^2\theta):
A=12×(3.0)2×2.0944A = \frac{1}{2} \times (3.0)^2 \times 2.0944
A=9.4248 units2A = 9.4248 \text{ units}^2
Note that θ\theta must be in radians when using these formulas. In degrees, the equivalents are s=θ360⋅2πrs = \frac{\theta}{360} \cdot 2\pi r and A=θ360⋅πr2A = \frac{\theta}{360} \cdot \pi r^2.
Interactive Sector and Arc VisualizerMax Bounds CircleSector Wedge AreaStart RadiusEnd RadiusArc Length HighlightCenter Pointr = 3.0s = 6.28A = 9.42
Sector Stats
s = 6.28A = 9.42

Trigonometric Functions - Theory & Concepts - Trigonometry Unit Circle

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

Trigonometric Functions - Theory & Concepts - Trigonometry Astc

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

Trigonometric Functions - Theory & Concepts - Trigonometry Trig Graphs

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

Trigonometric Identities - Theory & Concepts - Trigonometry Pythagorean Identity

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

Pythagorean Trigonometric Identity

Visualize the classic identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 geometrically. Adjust the angle to see how the squares scale while maintaining their sum.

Angle (θ\theta)36°
0°30°45°60°90°

Geometric Areas

Areaadj=cos⁡2θ\text{Area}_{\text{adj}} = \cos^2\theta0.8090^2 = 0.6545
Areaopp=sin⁡2θ\text{Area}_{\text{opp}} = \sin^2\theta0.5878^2 = 0.3455
Areahyp=12\text{Area}_{\text{hyp}} = 1^21.0000
Sum of Areas:0.6545 + 0.3455 = 1.0000
By the Pythagorean Theorem, in any right triangle with hypotenuse c=1c=1 and legs a=sin⁡θa=\sin\theta, b=cos⁡θb=\cos\theta:
a2+b2=c2  ⟹  sin⁡2θ+cos⁡2θ=1a^2 + b^2 = c^2 \implies \sin^2\theta + \cos^2\theta = 1
Pythagorean right triangle identity simulationHorizontal AxisVertical AxisUnit Circle Quarter ArcCosine Squared Square AreaSine Squared Square AreaHypotenuse Squared Square AreaRight TriangleRight Angle IndicatorTerminal Point on Unit Circlecos θsin θ1
Hypotenuse Square (Area = 1.00)

Trigonometric Identities - Theory & Concepts - Trigonometry Double Angle Sum

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

Angle Sum Geometric Proof

Explore the geometric proof of the identity sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta.

Angle α\alpha30°
10°45°
Angle β\beta25°
10°45°

Geometric Breakdown

Lower Segment (TR=QSTR = QS):
QS=sin⁡αcos⁡β=sin⁡(30∘)cos⁡(25∘)≈0.4532QS = \sin\alpha\cos\beta = \sin(30^\circ)\cos(25^\circ) \approx 0.4532
Upper Segment (PTPT):
PT=cos⁡αsin⁡β=cos⁡(30∘)sin⁡(25∘)≈0.3660PT = \cos\alpha\sin\beta = \cos(30^\circ)\sin(25^\circ) \approx 0.3660
Total Height (PR=sin⁡(α+β)PR = \sin(\alpha+\beta)):
sin⁡(30∘+25∘)=sin⁡(55∘)≈0.8192\sin(30^\circ+25^\circ) = \sin(55^\circ) \approx 0.8192
Sum of segments: 0.4532+0.3660=0.81920.4532 + 0.3660 = 0.8192
In right triangle △OPQ\triangle OPQ, hypotenuse OP=1OP = 1. Projection of PP onto the horizontal gives vertical height PR=sin⁡(α+β)PR = \sin(\alpha+\beta). This partition splits PRPR into PTPT and TRTR.
Geometric proof for double angle sum trigonometric identityX AxisY AxisUnit Circle BoundarySegment OQ (length = cos beta)Segment QP (length = sin beta)Segment OP (length = 1)Total Height PR (sin(alpha + beta))Vertical Height QS (cos beta * sin alpha)Horizontal line QTPoint OPoint PPoint QPoint RPoint SPoint TOPQRSTSegment TR (sin alpha * cos beta)Segment PT (cos alpha * sin beta)Angle alpha arcαAngle beta arcβAngle alpha at vertex Pα
PT = cos α · sin βTR = sin α · cos βPR = sin(α+β)

Trigonometric Identities - Theory & Concepts - Trigonometry Identity Verifier

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

Inverse Trigonometric Functions - Theory & Concepts - Trigonometry Inverse Trig Zone

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

Inverse Trigonometric Functions - Theory & Concepts - Trigonometry Inverse Composition

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

Inverse Trigonometric Functions - Theory & Concepts - Trigonometry Inverse Trig Graphs

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

Inverse Trig Graphs & Restrictions

Observe the reflection symmetry over the line y=xy = x and the crucial domain restriction required to define the inverse.

Value (xx)0.50
-101

Domain Restrictions

Base Function:f(x)=sin⁡(x)f(x) = \sin(x)
Restricted Domain of Base:[−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}](Highlights in thick blue on the graph)
Inverse Function:f−1(x)=arcsin⁡(x)f^{-1}(x) = \arcsin(x)
INVERSE DOMAIN:
[−1,1][-1, 1]
INVERSE RANGE:
[−π2,π2][-\frac{\pi}{2}, \frac{\pi}{2}]

Reflected Points (y=xy = x)

Inverse Point:(0.50, 0.52)
Base Point:(0.52, 0.50)
By reflecting any point (x,y)(x, y) on the inverse graph across the line y=xy = x, we get (y,x)(y, x)which lies exactly on the restricted base function's graph.
Overlay plotting of trigonometric function, inverse, and line of symmetry y = xHorizontal AxisVertical Axis-3-3-2-2-1-1112233Line y = xFull Base Trig WaveRestricted Domain Wave HighlightInverse Trig GraphReflection ConnectorInverse point (x, y)Base function point (y, x)
Base: sin(x)Inverse: arcsinSymmetry: y = x

Trigonometric Equations - Theory & Concepts - Trigonometry Trig Equations

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

Trigonometric Equations - Theory & Concepts - Trigonometry Periodic Solutions

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

Periodic Solutions Explorer

Solve sin⁡(x)=c\sin(x) = c and see how periodic solutions map onto the coordinate wave and the unit circle.

Value (cc)0.50
-1.00.01.0

Solutions in [−2π,2π][-2\pi, 2\pi]

x1x_{1}:
-5.7596 rad~ -1.83π
x2x_{2}:
-3.6652 rad~ -1.17π
x3x_{3}:
0.5236 rad~ 0.17π
x4x_{4}:
2.6180 rad~ 0.83π
Trigonometric functions are periodic. The equation sin⁡(x)=c\sin(x) = c has general solutions x=arcsin⁡(c)+2nπx = \arcsin(c) + 2n\pi and x=π−arcsin⁡(c)+2nπx = \pi - \arcsin(c) + 2n\pi where n∈Zn \in \mathbb{Z}.

Coordinate Wave: y = sin(x)

Periodic wave solutions plotHorizontal AxisVertical Axis-2π-1.5π-1π-0.5π0.5π1π1.5π2πTrigonometric WaveHorizontal Solution line y = cIntersection solution pointIntersection solution pointIntersection solution pointIntersection solution point

Unit Circle Projection

For sin⁡(x)=c\sin(x) = c, the horizontal line at height y=cy = c intersects the circle at the solution angles.

Periodic solution mapping on unit circleUnit Circle BoundaryProjection line y = cSolution terminal rayTerminal pointSolution terminal rayTerminal pointOrigin

Applications of Trigonometry - Theory & Concepts - Trigonometry Angle Elevation Depression

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

Angle of Elevation & Depression

Explore real-world right triangle applications by adjusting heights and observer distances.

Height (hh)50 m
10 m55 m100 m
Distance (dd)80 m
20 m85 m150 m

Triangle Results

Angle (θ\theta)
32.01°
0.5586 rad
Line of Sight (LL)
94.34 m
Hypotenuse
Step-by-Step Math

1. Find the Angle:

tan⁡(θ)=OppositeAdjacent=hd=5080=0.6250\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{d} = \frac{50}{80} = 0.6250θ=arctan⁡(0.6250)≈32.01∘\theta = \arctan(0.6250) \approx 32.01^\circ

2. Find Line of Sight (L):

L=h2+d2=502+802L = \sqrt{h^2 + d^2} = \sqrt{50^2 + 80^2}L=8900≈94.34 mL = \sqrt{8900} \approx 94.34\text{ m}
Angle of Elevation and Depression Right Triangle DiagramGround SurfaceTower Top BeaconHorizontal line of sight from tower topAngle of ElevationθAngle of Depressionθh = 50md = 80mL = 94.3m
Angle of Elevation (from Ground): 32.0°
Angle of Depression (from Top): 32.0°

Applications of Trigonometry - Theory & Concepts - Trigonometry Triangle Solver

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

Applications of Trigonometry - Theory & Concepts - Trigonometry Oblique Triangle

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

Spherical Trigonometry - Theory & Concepts - Trigonometry Spherical Triangle

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

Spherical Trigonometry - Theory & Concepts - Trigonometry Great Circle Distance

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

Great-Circle Distance Planner

Calculate distances and render the shortest spherical path (geodesic) between points.

Presets:
Point A (Red)
Latitude: 40.71°[-90°, 90°]
Longitude: -74.01°[-180°, 180°]
Point B (Blue)
Latitude: 51.51°[-90°, 90°]
Longitude: -0.13°[-180°, 180°]
Rotate Globe View
Distance Calculations
Central Angle (Δσ\Delta\sigma):50.09° (0.8743 rad)
Kilometers (km):5,570 km
Statute Miles:3,461 mi
Orthographic 3D Projection Globe and Great Circle PathThe Globe SphereShortest Path (Great-Circle Line)AB
Solid Line: Front Path (Visible)
Dashed Line: Back Path (Occluded)

Spherical Law of Cosines

For points with latitude ϕ1,ϕ2\phi_1, \phi_2 and difference in longitude Δλ\Delta\lambda, the central angle Δσ\Delta\sigma is:

cos⁡(Δσ)=sin⁡ϕ1sin⁡ϕ2+cos⁡ϕ1cos⁡ϕ2cos⁡(Δλ)\cos(\Delta\sigma) = \sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos(\Delta\lambda)
Applying Current Coordinates:
cos⁡(Δσ)=sin⁡(40.7∘)sin⁡(51.5∘)+cos⁡(40.7∘)cos⁡(51.5∘)cos⁡(73.9∘)\cos(\Delta\sigma) = \sin(40.7^\circ)\sin(51.5^\circ) + \cos(40.7^\circ)\cos(51.5^\circ)\cos(73.9^\circ)cos⁡(Δσ)=0.64153  ⟹  Δσ=0.87431 rad≈50.09∘\cos(\Delta\sigma) = 0.64153 \implies \Delta\sigma = 0.87431\text{ rad} \approx 50.09^\circDistance=R×Δσ=6371×0.87431≈5570.22 km\text{Distance} = R \times \Delta\sigma = 6371 \times 0.87431 \approx 5570.22\text{ km}

Complex Numbers and Polar Coordinates - Theory & Concepts - Trigonometry Polar Cartesian Converter

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

Coordinate System Converter

Toggle grids, adjust Cartesian or Polar coordinates, and verify trigonometric conversions.

X Coordinate6.0
-10.0010.0
Y Coordinate5.0
-10.0010.0
Coordinate Readings
CARTESIAN(6.00, 5.00)
POLAR(7.81, 40°)
Active Formulations
Cartesian → Polar:
r=x2+y2=(6.00)2+(5.00)2=7.81r = \sqrt{x^2 + y^2} = \sqrt{(6.00)^2 + (5.00)^2} = 7.81
θ=arctan⁡2(y,x)=arctan⁡2(5.00,6.00)≈39.8∘\theta = \arctan2(y, x) = \arctan2(5.00, 6.00) \approx 39.8^\circ
Polar → Cartesian:
x=rcos⁡θ=7.81cos⁡(39.8∘)=6.00x = r \cos\theta = 7.81 \cos(39.8^\circ) = 6.00
y=rsin⁡θ=7.81sin⁡(39.8∘)=5.00y = r \sin\theta = 7.81 \sin(39.8^\circ) = 5.00
Cartesian and Polar grids conversion diagramHorizontal Axis (X)Vertical Axis (Y)Origin (0,0) / PoleX (0°)Y (90°)x = 6.0y = 5.0r = 7.8θ
P = (6.00, 5.00)
r = 7.81, θ = 39.8°

Complex Numbers and Polar Coordinates - Theory & Concepts - Trigonometry Complex Roots

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

Hyperbolic Trigonometry - Theory & Concepts - Trigonometry Hyperbolic Functions

Comprehensive study of Hyperbolic Sine, Cosine, Tangent, and their relationships to the hyperbola and exponential functions.

Circular vs. Hyperbolic Functions

Compare the unit circle x2+y2=1x^2+y^2=1 and the unit hyperbola x2−y2=1x^2-y^2=1. Observe how the parameters represent sector areas.

1. The Unit Circle

Equation: x2+y2=1x^2 + y^2 = 1
Angle / Parameter (tt)1.00 rad (57°)
0.00π (~3.14)2π (~6.28)
COORDINATES(0.540, 0.841)
SECTOR AREAA = 0.500
x=cos⁡(t)=0.540x = \cos(t) = 0.540y=sin⁡(t)=0.841y = \sin(t) = 0.841
Circular functions trigonometric visualization on a unit circleP(cos t, sin t)

2. The Unit Hyperbola

Equation: x2−y2=1x^2 - y^2 = 1
Hyperbolic Parameter (uu)1.00
-2.0002.00
COORDINATES(1.543, 1.175)
SECTOR AREAA = 0.500
x=cosh⁡(u)=1.543x = \cosh(u) = 1.543y=sinh⁡(u)=1.175y = \sinh(u) = 1.175
Hyperbolic functions visualization on a unit hyperbola

Exponential Formulations for Hyperbolic Functions

While circular functions are defined using trigonometric angles on a circle, hyperbolic functions are defined algebraically using exponents eue^u and e−ue^{-u}. Geometrically, both represent twice the area of their respective sectors.

Hyperbolic Cosine:
cosh⁡(u)=eu+e−u2\cosh(u) = \frac{e^u + e^{-u}}{2}
Value: cosh⁡(1.00)=2.718+0.3682=1.5431\cosh(1.00) = \frac{2.718 + 0.368}{2} = 1.5431
Hyperbolic Sine:
sinh⁡(u)=eu−e−u2\sinh(u) = \frac{e^u - e^{-u}}{2}
Value: sinh⁡(1.00)=2.718−0.3682=1.1752\sinh(1.00) = \frac{2.718 - 0.368}{2} = 1.1752
Hyperbolic Tangent:
tanh⁡(u)=sinh⁡(u)cosh⁡(u)=eu−e−ueu+e−u\tanh(u) = \frac{\sinh(u)}{\cosh(u)} = \frac{e^u - e^{-u}}{e^u + e^{-u}}
Value: tanh⁡(1.00)=1.1751.543=0.7616\tanh(1.00) = \frac{1.175}{1.543} = 0.7616

Hyperbolic Trigonometry - Theory & Concepts - Trigonometry Hyperbolic Identities

Comprehensive study of Hyperbolic Sine, Cosine, Tangent, and their relationships to the hyperbola and exponential functions.

Hyperbolic Identity Verifier

Choose an identity, modify inputs, and verify algebraic equality.

Variable (xx)1.2
-2.50.02.5
Identity Verification
LHS Value
cosh⁡2(x)−sinh⁡2(x)=1.00000\cosh^2(x) - \sinh^2(x) = 1.00000
RHS Value
1=1.000001 = 1.00000
Absolute Difference:0.00000000
Calculated Steps
Left Hand Side:cosh⁡2(1.2)−sinh⁡2(1.2)=(1.8107)2−(1.5095)2\cosh^2(1.2) - \sinh^2(1.2) = (1.8107)^2 - (1.5095)^2
Right Hand Side:3.2785−2.2785=1.00003.2785 - 2.2785 = 1.0000
Plot of hyperbolic functions overlay for identity validationHorizontal Axis (t)Vertical Axis (y)-3-2-11232468y = cosh^2(t)y = sinh^2(t)y = cosh^2(t) - sinh^2(t) = 1x = 1.2y = cosh²ty = sinh²tDifference = 1
Because mathematical identities are true for all values of the variable domain, the plotted Left Hand Side (LHS) curve and Right Hand Side (RHS) curve lie perfectly on top of each other. Contrast this with standard algebraic equations, which are only true for specific solutions.