Trigonometry

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 Formulas

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.

Sine Function (sin)

\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} = \frac{1}{\sin \theta}

Secant Function (sec)

\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} = \frac{1}{\cos \theta}

Cotangent Function (cot)

\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} = \frac{1}{\tan \theta}

Secant Reciprocal Identity

\sec \theta = \frac{1}{\cos \theta}

Cosecant Reciprocal Identity

\csc \theta = \frac{1}{\sin \theta}

Cotangent Reciprocal Identity

\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

\sin^2 \theta + \cos^2 \theta = 1

Tangent Pythagorean Identity

1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

\cos(-\theta) = \cos \theta

Secant Even Identity

\sec(-\theta) = \sec \theta

Sine Odd Identity

\sin(-\theta) = -\sin \theta

Tangent Odd Identity

\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

\cot(-\theta) = -\cot \theta

Sine Function (sin)

\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite Side}} = \frac{1}{\sin \theta}

Secant Function (sec)

\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} = \frac{1}{\cos \theta}

Cotangent Function (cot)

\cot \theta = \frac{\text{Adjacent Side}}{\text{Opposite Side}} = \frac{1}{\tan \theta}

Secant Reciprocal Identity

\sec \theta = \frac{1}{\cos \theta}

Cosecant Reciprocal Identity

\csc \theta = \frac{1}{\sin \theta}

Cotangent Reciprocal Identity

\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

\sin^2 \theta + \cos^2 \theta = 1

Tangent Pythagorean Identity

1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

\cos(-\theta) = \cos \theta

Secant Even Identity

\sec(-\theta) = \sec \theta

Sine Odd Identity

\sin(-\theta) = -\sin \theta

Tangent Odd Identity

\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

\cot(-\theta) = -\cot \theta

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Trigonometry Terms and Definitions

Trigonometry

The branch of mathematics that examines the relationships between the angles and sides of triangles, primarily through trigonometric functions like sine, cosine, and tangent.

Plane angle

A geometric figure formed by two rays sharing a common endpoint, measured as the rotation from one ray (the initial side) to the other (the terminal side).

Vertex

The common endpoint of two rays or line segments that define an angle or form a corner in a geometric figure.

Initial side

The fixed starting ray of an angle from which rotation begins to define the angle.

Terminal side

The ray that results after rotating the initial side to form an angle.

Positive angle

An angle measured counterclockwise from the initial side to the terminal side.

Negative angle

An angle measured clockwise from the initial side to the terminal side.

Degree

A unit of angular measure where one degree equals 1/360 of a complete circle.

Minute

A subunit of a degree, where one minute equals 1/60 of a degree.

Second

A subunit of a minute, where one second equals 1/60 of a minute or 1/3600 of a degree.

Radian

A measure of an angle based on the arc length of a circle, defined as the angle subtended by an arc equal in length to the radius of the circle.

Arc length

The distance along the curve of a circle, determined by the product of the circle's radius and the central angle in radians.

Central angle

An angle formed by two radii of a circle, with its vertex at the center of the circle.

Unit circle

A circle with a radius of one unit, used in trigonometry to define and evaluate trigonometric functions.

Sector

A portion of a circle bounded by two radii and the arc between them.

Area of a sector

The area of a slice of a circle, calculated as half the product of the radius squared and the central angle in radians.

Angular velocity

The rate of change of an angle over time, expressed in radians per unit time.

Linear velocity

The rate at which a point moves along the circumference of a circular path, calculated as the product of the radius and the angular velocity.

Directed line

A line with an assigned positive and negative direction, used to measure distances with sign.

Rectangular coordinate system

A two-dimensional plane defined by perpendicular horizontal and vertical axes intersecting at an origin, used to locate points using ordered pairs of numbers.

Abscissa

The horizontal coordinate of a point in a rectangular coordinate system, representing its distance from the vertical axis.

Ordinate

The vertical coordinate of a point in a rectangular coordinate system, representing its distance from the horizontal axis.

Quadrant

One of the four regions of a rectangular coordinate system, divided by the x-axis and y-axis.

Radius vector

The straight-line distance from the origin to a point in a rectangular coordinate system.

Angle in standard position

An angle with its vertex at the origin of a coordinate system and its initial side along the positive x-axis.

Coterminal angles

Angles that share the same terminal side when placed in standard position, differing by integer multiples of 360°.

Quadrantal angles

Angles whose terminal sides lie along the axes in a coordinate system, such as 0°, 90°, 180°, and 270°.

Reciprocal trigonometric functions

Functions defined as the reciprocals of the primary trigonometric functions, including cosecant (csc), secant (sec), and cotangent (cot).

Unit circle

A circle with a radius of one, centered at the origin in a coordinate plane, used to define trigonometric functions.

Circular functions

Functions like sine and cosine defined as the coordinates of a point on the unit circle, applicable to all real numbers.

Undefined trigonometric functions

Trigonometric functions that do not have a value for specific angles due to division by zero.

Arc length on unit circle

The length of an arc on the unit circle, numerically equal to the measure of the angle in radians.

Wrapping function

A function that maps real numbers to points on the unit circle by associating arc lengths with coordinates (cos(s), sin(s)).

Signs of trigonometric functions

The positivity or negativity of trigonometric functions in different quadrants based on the signs of x and y coordinates.

Circular angle measures

Angles measured in radians, where the arc length on the unit circle equals the angle in radians.

Domain of trigonometric functions

The set of input values (angles) for which trigonometric functions are defined.

Evaluation of trigonometric functions

Calculating specific trigonometric values for given angles using known properties, quadrants, or the unit circle.

Periodicity of trigonometric functions

A property of trigonometric functions where their values repeat at regular intervals, such as 2π for sine and cosine.

Even function

A function f(x) that satisfies f(−x) = f(x), with the graph symmetric about the y-axis; examples include cosine and secant.

Odd function

A function $f(x)$ that satisfies $f(−x) = −f(x)$, with the graph symmetric about the origin; examples include sine and tangent.

Trigonometry

The branch of mathematics that examines the relationships between the angles and sides of triangles, primarily through trigonometric functions like sine, cosine, and tangent.

Plane angle

A geometric figure formed by two rays sharing a common endpoint, measured as the rotation from one ray (the initial side) to the other (the terminal side).

Vertex

The common endpoint of two rays or line segments that define an angle or form a corner in a geometric figure.

Initial side

The fixed starting ray of an angle from which rotation begins to define the angle.

Terminal side

The ray that results after rotating the initial side to form an angle.

Positive angle

An angle measured counterclockwise from the initial side to the terminal side.

Negative angle

An angle measured clockwise from the initial side to the terminal side.

Degree

A unit of angular measure where one degree equals 1/360 of a complete circle.

Minute

A subunit of a degree, where one minute equals 1/60 of a degree.

Second

A subunit of a minute, where one second equals 1/60 of a minute or 1/3600 of a degree.

Radian

A measure of an angle based on the arc length of a circle, defined as the angle subtended by an arc equal in length to the radius of the circle.

Arc length

The distance along the curve of a circle, determined by the product of the circle's radius and the central angle in radians.

Central angle

An angle formed by two radii of a circle, with its vertex at the center of the circle.

Unit circle

A circle with a radius of one unit, used in trigonometry to define and evaluate trigonometric functions.

Sector

A portion of a circle bounded by two radii and the arc between them.

Area of a sector

The area of a slice of a circle, calculated as half the product of the radius squared and the central angle in radians.

Angular velocity

The rate of change of an angle over time, expressed in radians per unit time.

Linear velocity

The rate at which a point moves along the circumference of a circular path, calculated as the product of the radius and the angular velocity.

Directed line

A line with an assigned positive and negative direction, used to measure distances with sign.

Rectangular coordinate system

A two-dimensional plane defined by perpendicular horizontal and vertical axes intersecting at an origin, used to locate points using ordered pairs of numbers.

Abscissa

The horizontal coordinate of a point in a rectangular coordinate system, representing its distance from the vertical axis.

Ordinate

The vertical coordinate of a point in a rectangular coordinate system, representing its distance from the horizontal axis.

Quadrant

One of the four regions of a rectangular coordinate system, divided by the x-axis and y-axis.

Radius vector

The straight-line distance from the origin to a point in a rectangular coordinate system.

Angle in standard position

An angle with its vertex at the origin of a coordinate system and its initial side along the positive x-axis.

Coterminal angles

Angles that share the same terminal side when placed in standard position, differing by integer multiples of 360°.

Quadrantal angles

Angles whose terminal sides lie along the axes in a coordinate system, such as 0°, 90°, 180°, and 270°.

Reciprocal trigonometric functions

Functions defined as the reciprocals of the primary trigonometric functions, including cosecant (csc), secant (sec), and cotangent (cot).

Unit circle

A circle with a radius of one, centered at the origin in a coordinate plane, used to define trigonometric functions.

Circular functions

Functions like sine and cosine defined as the coordinates of a point on the unit circle, applicable to all real numbers.

Undefined trigonometric functions

Trigonometric functions that do not have a value for specific angles due to division by zero.

Arc length on unit circle

The length of an arc on the unit circle, numerically equal to the measure of the angle in radians.

Wrapping function

A function that maps real numbers to points on the unit circle by associating arc lengths with coordinates (cos(s), sin(s)).

Signs of trigonometric functions

The positivity or negativity of trigonometric functions in different quadrants based on the signs of x and y coordinates.

Circular angle measures

Angles measured in radians, where the arc length on the unit circle equals the angle in radians.

Domain of trigonometric functions

The set of input values (angles) for which trigonometric functions are defined.

Evaluation of trigonometric functions

Calculating specific trigonometric values for given angles using known properties, quadrants, or the unit circle.

Periodicity of trigonometric functions

A property of trigonometric functions where their values repeat at regular intervals, such as 2π for sine and cosine.

Even function

A function f(x) that satisfies f(−x) = f(x), with the graph symmetric about the y-axis; examples include cosine and secant.

Odd function

A function $f(x)$ that satisfies $f(−x) = −f(x)$, with the graph symmetric about the origin; examples include sine and tangent.

See All 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.

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Trigonometric Identities

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(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

\csc(\theta) =\frac{1}{\sin(\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, including reflections across axes and rotations. For instance,

\cos(-\theta) = \cos(\theta)

showcases cosine’s evenness, while

\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)

exemplifies co-function symmetry.

The Pythagorean identities, such as

\sin^2(\theta) + \cos^2(\theta) = 1

, are central to the structure and serve as a gateway to more complex relationships.

Angle sum and difference formulas—like

\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)

—are derived through coordinate geometry or rotation matrices, enabling the construction of double-angle, half-angle, and triple-angle identities, such as

\cos(2\theta) = 2\cos^2(\theta) - 1

.

Product-to-sum identities, for example

\sin(a)\sin(b) = \frac{1}{2}[\cos(a - b) - \cos(a + b)]

, reorganize these angle relationships for simplification in both theory and applications.

Inverse trigonometric identities like

\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}

, along with hyperbolic analogs and Euler’s identity

e^{i\theta} = \cos(\theta) + i\sin(\theta)

, extend trigonometry into broader mathematical contexts. Throughout this structure, many identities emerge as special cases of these general forms, demonstrating a coherent and logical progression.

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Trigonometric Functions

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. On the child page, we will examine:

* 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

While more specialized trigonometric forms (e.g., inverse and hyperbolic functions) are acknowledged, they are treated in their own contexts where relevant.

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

Unit Circle

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 Unit Circle page, we explore this model in detail using an interactive 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.

Explore the page to see how angles, coordinates, and functions all come together in one unified model.

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Trigonometry Symbols Reference

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, Euler's formula, and important sum and difference identities—all presented with precise LaTeX code for academic writing and mathematical typesetting.

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Tools

Degree Radians Angle Converter

Convert between Degrees to Radians and back with our visual intuitive and interactive converter

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