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Home › Trigonometry
SECTIONTrigonometry17 subsectionsBROWSE ALL ↓INTERACTIVE1 TOOLTrigonometryVisual Tools11 toolsJump to ↓fREFERENCE101 ITEMSfTrigonometryFormulas andIdentities68 itemsJump to ↓AaTrigonometry Termsand Definitions33 itemsJump to ↓§CORE TOPICS14 SUBSECTIONSAnglesJump to ↓Degrees and RadiansJump to ↓Trigonometric EquationsJump to ↓Trigonometric FunctionsJump to ↓Trigonometric GraphsJump to ↓Trigonometric IdentitiesJump to ↓Trigonometric InequalitiesJump to ↓Inverse TrigonometricFunctionsJump to ↓Trigonometric PropertiesJump to ↓Right Triangle TrigonometryJump to ↓Law of Sines and Law ofCosinesJump to ↓Unit CircleJump to ↓Interactive ToolsJump to ↓Trigonometry SymbolsReferenceJump to ↓

Introduction to Trigonometry

While mainly focusing on relationships between the angles and sides of triangles, trigonometry extends beyond simple geometric shapes, offering tools to describe patterns involving periodicity and circular motion. This makes trigonometry essential in fields ranging from physics and engineering to computer graphics and navigation.

Here’s what students can expect to learn:

1. Core Concepts:
• Angles and Their Measures: Degrees, radians, and how they relate to circles.
• Trigonometric Functions: Sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent).
• Unit Circle: A key tool for understanding how trigonometric functions behave across all angles.
• Inverse Trigonometric Functions: Solving equations to find angles.

2. Applications:
• Solving Triangles: Using the laws of sines and cosines to find unknown sides or angles.
• Wave Behavior: Modeling oscillations in physics, sound waves, and light waves.
• Circular and Periodic Motion: Understanding orbits, pendulums, and cycles.

3. Advanced Topics:
• Identities: Simplifying expressions and solving equations using formulas like the Pythagorean identity, angle addition formulas, and double-angle formulas.
• Graphs of Trigonometric Functions: Visualizing periodic patterns and their transformations (shifts, stretches, reflections).
• Complex Numbers and Euler’s Formula: Exploring the deep connection between trigonometry and algebra.

4. Skills Developed:
• Problem-solving: Tackling real-world problems involving geometry, motion, and oscillation.
• Visualization: Interpreting and sketching graphs and diagrams to represent abstract relationships.
• Logical reasoning: Manipulating trigonometric expressions and proofs.

Trigonometry teaches students how to connect abstract mathematical concepts with practical scenarios, developing versatile skills useful in scientific modeling, technical fields, and everyday problem-solving.

Trigonometry Visual Tools

11 toolsExplore Trigonometry Visual Tools →
Each tool turns a piece of trigonometry into something you can manipulate, watch, and step through — angles, the unit circle, graph transformations, identity proofs, triangle solving, and inverse functions.
45°special angles
degrad42°
Visual Tool

Angle Explorer

An interactive angle visualizer that lets you set any angle in degrees or radians and instantly see its type, quadrant, reference angle, and its complementary, supplementary, and coterminal partners. Toggle reference lines, jump to preset unit-circle angles, and read live values for sin, cos, tan, csc, sec, and cot, including exact special-angle forms.

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Visual Tool

Interactive Angle Types Explorer

An interactive hub for nine core angle concepts in trigonometry. Drag handles or use presets to explore basic classifications (acute, right, obtuse, straight, reflex), pairwise relationships (complementary, supplementary, vertical, adjacent), and trigonometric foundations (standard position, reference angles, coterminal angles, the 16 special unit-circle angles, and directed angles with even and odd verification).

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cossinθ
Visual Tool

Basic Trigonometric Identities Explorer

Drag the blue dot around the unit circle to watch sine, cosine, tangent, and their reciprocals emerge as signed legs and ratios. Switch between the six functions via tabs, step through a five-stage derivation for each, and use the multi-turn slider to explore periodicity and coterminal angles.

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θ2θsin 2θ
Visual Tool

Double Angle Identities Explorer

Double-Angle identities explorer covering all six trigonometric functions. Animated geometric proofs derive sin(2θ) and cos(2θ) from a bisected isosceles triangle, while algebraic derivation cards handle tan(2θ), csc(2θ), sec(2θ), and cot(2θ) with one-click jumps back to their source proofs. A live formula table and verification cards confirm each identity numerically as you sweep the angle.

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sincos
Visual Tool

Interactive Trigonometric Functions Graphs

An interactive trigonometric functions visualizer that graphs sine, cosine, tangent, cosecant, secant, and cotangent in real time. Switch between degrees and radians, drag the angle slider through a full rotation, jump to special-angle presets, and watch live function values, periods, ranges, and asymptotes update on the plot and in the explanation panel.

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ASTC
Visual Tool

Interactive Trigonometric Functions Signs by Quadrants

An interactive quadrant explorer that shows the sign of every trigonometric function in each of the four quadrants. Click a quadrant or press 1-4, pick a function, and watch the unit circle, coordinate signs, ASTC rule, and step-by-step reasoning update together. A clean way to internalize why sine, cosine, tangent, and their reciprocals flip sign as the angle moves around the circle.

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αα/2sin(α/2)
Visual Tool

Half Angle Identities Explorer

Half-Angle identities explorer covering all six trigonometric functions. Animated geometric proofs derive sin(α/2) and cos(α/2) from a bisected isosceles triangle, while algebraic derivation cards handle tan(α/2), csc(α/2), sec(α/2), and cot(α/2) with one-click jumps back to their source proofs. A live formula table and verification cards confirm each identity numerically as you sweep α from 20° to 160°.

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θ−θsin(−θ) = −sin θ
Visual Tool

Negative Angle Identities Explorer

Negative-Angle identities explorer covering all six trigonometric functions. Animated reflection proofs derive sin(-θ) = -sin θ and cos(-θ) = cos θ from a point and its mirror image across the x-axis on the unit circle, while algebraic derivation cards handle tan, csc, sec, and cot. A live formula table and verification cards label each function

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1sin²θ + cos²θ = 1
Visual Tool

Pythagorean Identities Explorer

Pythagorean identities explorer covering all six trigonometric functions. The base identity sin²θ + cos²θ = 1 is proved geometrically inside the unit circle, then tan, csc, sec, and cot forms follow by dividing through and rearranging. Animated steps, algebraic derivation cards with jump-to-source buttons, and a live formula table verify every identity numerically as you sweep θ through the first quadrant.

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θπ−θsin(π−θ) = sin θ
Visual Tool

Supplementary Angle Identities Visualizer

Explore all six trig functions at π − θ across two complementary views. Sin and cos appear as reflections across the y-axis with live unit-circle geometry; tan, csc, sec, and cot derive algebraically from those two roots. Step through proofs, switch tabs, and verify each identity numerically at any angle.

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cab
Visual Tool

Interactive Triangle Explorer

An interactive triangle visualizer with twelve preloaded scenarios covering classifications (equilateral, isosceles, acute, obtuse, right scalene), special right triangles (45-45-90, 30-60-90, 3-4-5, 5-12-13), and the laws of sines and cosines. Drag vertices, lock up to two angles, zoom freely, and read live stats and context-aware explanations including Pythagoras, SOH CAH TOA, and the trigonometric laws.

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Explore Trigonometry Visual Tools →

Trigonometry Formulas and Identities

68 itemsSee All Trigonometry Formulas and Identities →
The Trigonometry Formulas page covers essential identities and relationships involving angles, ratios, and periodic functions. It includes fundamental formulas such as Pythagorean identities, angle sum and difference identities, double and half-angle formulas, product-to-sum transformations, and laws of sines and cosines. Each formula is presented with definitions, usage notes, and step-by-step examples for solving geometric and algebraic problems.
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Trigonometry Terms and Definitions

33 itemsSee All Trigonometry Terms and Definitions →
The Trigonometry Terms and Definitions page offers a comprehensive glossary of key concepts in trigonometry, organized by categories such as Angles, Triangles, Trigonometric Functions, Identities, Graphs, and Equations. From foundational terms like sine and cosine to advanced concepts like amplitude modulation and inverse identities, each entry is clearly defined to support a deeper understanding of trigonometric principles and their mathematical applications.
Angles & Measurement
Angle — a figure formed by two rays sharing a common endpoint, measuring rotation
Degree — 1/360 of a full rotation
Radian — the angle subtended by an arc equal in length to the radius
Arc Length — distance along a circular arc: s = rθ
Central Angle — an angle whose vertex is at the center of a circle
Sector — the region between two radii and their intercepted arc
Unit Circle — the circle x2 + y2 = 1 centered at the origin
Initial Side — the fixed ray from which rotation begins
Terminal Side — the ray obtained after rotation
Positive Angle — counterclockwise rotation
Negative Angle — clockwise rotation
Standard Position — vertex at origin, initial side along the positive x-axis
Coterminal Angles — angles sharing the same terminal side
Quadrantal Angles — angles whose terminal side lies on an axis
Reference Angle — the acute angle between the terminal side and the x-axis
Complementary Angles — two angles summing to 90°
Supplementary Angles — two angles summing to 180°
Functions
Sine — the y-coordinate on the unit circle
Cosine — the x-coordinate on the unit circle
Tangent — the ratio sin θ / cos θ, the slope of the terminal side
Cosecant — reciprocal of sine
Secant — reciprocal of cosine
Cotangent — reciprocal of tangent
Trigonometric Ratio — a ratio of two sides of a right triangle relative to an acute angle
Periodic Function — a function that repeats at regular intervals
Inverse Trigonometric Function — returns the angle for a given trigonometric value
Right Triangle
Hypotenuse — the side opposite the right angle
Adjacent Side — the leg next to the chosen acute angle
Opposite Side — the leg across from the chosen acute angle
Graphs
Amplitude — maximum distance from midline to peak: |A|
Period — horizontal length of one cycle: 2π/|B|
Phase Shift — horizontal displacement: C/B
Frequency — cycles per unit interval: 1/T
Angles & Measurement17
AngleA figure formed by two rays sharing a common endpoint (vertex), representing the amount and direction of rotation from one ray to the other.Read more →Initial SideThe fixed ray from which an angle's rotation begins, lying along the positive xx-axis when the angle is in standard position.Read more →Terminal SideThe ray obtained by rotating the initial side through the given angle; its position determines all trigonometric function values.Read more →Positive AngleAn angle generated by counterclockwise rotation from the initial side to the terminal side.Read more →Negative AngleAn angle generated by clockwise rotation from the initial side to the terminal side.Read more →DegreeA unit of angle measurement equal to 1360\frac{1}{360} of a full rotation, denoted by the symbol °°.Read more →RadianThe angle subtended at the center of a circle by an arc whose length equals the radius: θ=sr\theta = \frac{s}{r}.Read more →Arc LengthThe distance along a circular arc intercepted by a central angle: s=rθs = r\theta, where θ\theta is in radians.Read more →Central AngleAn angle whose vertex is at the center of a circle and whose sides are radii intercepting an arc.Read more →Unit CircleThe circle of radius 11 centered at the origin, defined by x2+y2=1x^2 + y^2 = 1, whose points encode trigonometric values as coordinates.Read more →SectorThe region enclosed by two radii and the arc between them, with area A=12r2θA = \frac{1}{2}r^2\theta where θ\theta is in radians.Read more →Angle in Standard PositionAn angle placed on the coordinate plane with its vertex at the origin and its initial side along the positive xx-axis.Read more →Coterminal AnglesTwo angles that share the same terminal side when placed in standard position, differing by an integer multiple of 360°360° (or 2π2\pi).Read more →Quadrantal AnglesAngles whose terminal side lies along a coordinate axis: 0°0°, 90°90°, 180°180°, 270°270°, and their coterminal equivalents.Read more →Reference AngleThe acute angle between the terminal side of θ\theta and the xx-axis, always in [0°,90°][0°, 90°] (or [0,π2][0, \frac{\pi}{2}]).Read more →Complementary AnglesTwo angles whose measures sum to 90°90° (or π2\frac{\pi}{2} radians).Read more →Supplementary AnglesTwo angles whose measures sum to 180°180° (or π\pi radians).Read more →
SineThe trigonometric function that maps an angle θ\theta to the yy-coordinate of the corresponding point on the unit circle: sin⁡θ=y\sin\theta = y.Read more →CosineThe trigonometric function that maps an angle θ\theta to the xx-coordinate of the corresponding point on the unit circle: cos⁡θ=x\cos\theta = x.Read more →TangentThe ratio of sine to cosine: tan⁡θ=sin⁡θcos⁡θ=yx\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}, geometrically the slope of the terminal side.Read more →CosecantThe reciprocal of sine: csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}.Read more →SecantThe reciprocal of cosine: sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}.Read more →CotangentThe reciprocal of tangent, equivalently the ratio of cosine to sine: cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}.Read more →Trigonometric RatioA ratio of two sides of a right triangle relative to one of its acute angles, defining the six trigonometric functions geometrically.Read more →Periodic FunctionA function ff for which there exists a positive constant TT such that f(x+T)=f(x)f(x + T) = f(x) for all xx in the domain. The smallest such TT is the…Read more →Inverse Trigonometric FunctionA function that reverses a trigonometric function on a restricted domain, returning the angle whose trigonometric value is the given input.Read more →
View All Trigonometry Terms and Definitions →

Angles

In some way, we can see an angle as a mathematical model that represents the amount of rotation between two rays sharing a common endpoint. It is one of the central concepts in trigonometry, alongside the right triangle and the unit circle. Angles provide the basis for defining trigonometric functions, which depend on rotational input to describe relationships between lengths, positions, and periodic behavior. Through this role, the angle becomes essential for understanding how these functions behave, both algebraically and geometrically.

Measurement of angles is defined through two principal units — degrees and radians — and includes distinctions between positive and negative directionality, as well as values extending beyond a single rotation. The concept of coterminal angles arises naturally within this extended framework.

Angular magnitudes are categorized based on their size: zero, acute, right, obtuse, straight, reflex, and full angles. These classifications serve as a basis for identifying structural properties in figures and for formulating general principles in planar analysis.

Relationships between angles — such as complementarity, supplementarity, adjacency, and opposition — are defined by positional constraints and linear combinations.

In geometric contexts, angles determine the internal structure of triangles and polygons, govern the behavior of circular segments, and appear in the formulation of central and inscribed arc measures. Within trigonometry, angles are treated as rotational parameters on the unit circle, where quadrant location and reference angles determine function values and signs. Standard angles play a central role in simplifying expressions and solving equations.

These definitions extend to the behavior of trigonometric functions across their domains, including periodicity, symmetry, and transformation. The treatment also includes inverse functions and the role of angles in modeling cyclic and rotational phenomena in applied settings. The topic as a whole integrates measurement, classification, and function, providing a complete mathematical foundation for angular analysis.

Degrees and Radians

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Learn degree and radian angle measurement, convert between the two systems, and apply radians to arc length and sector area formulas with clear explanations.
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Trigonometric Equations

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Learn to solve trigonometric equations: isolate the function, find reference angles, generate all solutions, and handle quadratic and multiple-angle equations.
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Trigonometric Functions

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Trigonometric functions lie at the heart of trigonometry, originally emerging from the study of right triangles and the unit circle. Today, they serve as indispensable tools in mathematics, especially in calculus, analysis, and mathematical modeling of periodic phenomena.

The most fundamental functions are the sine, cosine, and tangent, along with their reciprocals cosecant, secant, and cotangent. These form the core set of basic trigonometric functions. Other function families — such as inverse, hyperbolic, and complex extensions — also exist and are important in advanced contexts, but this section focuses primarily on the classical real-valued functions and their mathematical behavior.

This part of the trigonometry module is dedicated to exploring the analytical structure of trigonometric functions:

* Definitions: from geometric constructions to analytic formulations via power series and differential equations
* Properties: including periodicity, symmetry, continuity, and boundedness
* Identities: such as angle sum/difference formulas, double-angle identities, and Pythagorean relations
* Graphs and Transformations: visual behaviors and effects of shifts, stretches, and reflections
* Equations: solving trigonometric equations and analyzing their solutions
* Mathematical Applications: their roles in Fourier analysis, differential equations, and linear algebra

Whether you are analyzing waveforms, studying rotations, or decomposing functions into periodic components, a strong grasp of trigonometric functions provides a powerful mathematical toolkit.
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Trigonometric Graphs

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Learn to graph sine, cosine, tangent, and reciprocal functions. Master amplitude, period, phase shift, vertical shift, and the key-point method for graphing.
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Trigonometric Identities

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Trigonometric identities form a rich and interconnected system rooted in geometric definitions and algebraic transformations. At the foundation lie the reciprocal and quotient identities, such as tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, that emerge directly from the unit circle and right-triangle definitions of the trigonometric functions.

Building upon this base are the symmetry-based identities — such as the even-odd and co-function properties — which reflect the inherent symmetries of the unit circle. For instance, cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta) showcases the evenness of cosine, while sin⁡(π2−θ)=cos⁡(θ)\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) exemplifies co-function symmetry.

The Pythagorean identities, such as sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1, are central to the structure. Angle sum and difference formulas — like sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) — enable the construction of double-angle, half-angle, and triple-angle identities, such as cos⁡(2θ)=2cos⁡2(θ)−1\cos(2\theta) = 2\cos^2(\theta) - 1.

Product-to-sum identities reorganize these angle relationships for simplification, while sum-to-product identities transform sums and differences into products, enabling alternative approaches to complex expressions.

Inverse trigonometric identities like sin⁡−1(x)+cos⁡−1(x)=π2\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}, along with hyperbolic analogs and the Euler identity eiθ=cos⁡(θ)+isin⁡(θ)e^{i\theta} = \cos(\theta) + i\sin(\theta), extend trigonometry into broader mathematical contexts.
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Trigonometric Inequalities

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Master trigonometric inequalities: solve using graphs and the unit circle, express solutions in interval notation, and handle compound inequalities.
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Inverse Trigonometric Functions

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Learn inverse trigonometric functions: arcsine, arccosine, arctangent, their restricted domains, ranges, graphs, and how to evaluate compositions.
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Trigonometric Properties

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Explore the key properties of trigonometric functions: periodicity, even/odd symmetry, boundedness, zeros, continuity, monotonicity, and extrema explained.
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Right Triangle Trigonometry

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Master right triangle trigonometry: SOH CAH TOA ratios, finding missing sides and angles, 45-45-90 and 30-60-90 special triangles, and real-world applications.
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Law of Sines and Law of Cosines

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Master the Law of Sines and Law of Cosines for oblique triangles: the ambiguous case, choosing the right law, triangle area, and Heron
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Unit Circle

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The unit circle is one of the most powerful conceptual tools in trigonometry. It transforms the study of triangles into the study of circular motion, periodic behavior, and coordinate geometry — all within a single unified framework. By fixing a circle of radius 1 at the origin of the coordinate plane, we gain a clean, visual model that defines sine, cosine, and tangent not just for acute angles, but for all real numbers and even complex values.

On the interactive Unit Circle visual tool, we explore this model in detail with a visualizer that allows users to trace angles, visualize coordinate projections, and see how trigonometric functions behave dynamically across all four quadrants. The unit circle demystifies why certain angles produce clean values, explains sign changes through quadrants, and shows how right triangle ratios extend naturally into the full coordinate plane.

Key topics include:

* The definition and structure of the unit circle
* Why the radius is set to 1 (and how that simplifies trigonometric ratios)
* Angle measurement systems (degrees vs. radians)
* The role of quadrants in determining function signs
* The importance of special angles (like 30°, 45°, and 60°)

More than just a diagram, the unit circle acts as a conceptual bridge between geometry, algebra, and analysis. It provides the foundation for defining trigonometric functions analytically, extending them to calculus, complex numbers, and Fourier theory. If you want to truly understand trigonometry — not just memorize formulas — the unit circle is the place to start.
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Interactive Tools

Converter
Degree ↔ Radians Converter
Convert between degrees and radians with a visual, intuitive, and interactive converter.
Open tool → Visual Tool
Unit Circle Explorer
Explore the unit circle for a better understanding of the main trigonometry concepts.
Open tool → Calculator
Trigonometry Calculator
Calculate the values of all main trigonometric functions for angles measured in degrees and radians.
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Trigonometry Symbols Reference

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Our Trigonometry Symbols page offers a comprehensive collection of notation used in trigonometric mathematics. This reference serves as a valuable resource for students and educators working with angular relationships and periodic functions.

The guide organizes symbols into functional categories including basic trigonometric functions (sin(θ), cos(θ), tan(θ)), their inverse functions (sin⁻¹(x), cos⁻¹(x)), and fundamental identities such as the Pythagorean identity sin²(θ) + cos²(θ) = 1. It extends to practical applications like the Law of Sines and Cosines for triangle calculations, unit circle relationships, and hyperbolic functions.

Advanced sections cover complex number representations using trigonometric forms, the Euler formula, and important sum and difference identities — all presented with precise LaTeX code for academic writing and mathematical typesetting.

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