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Trigonometry

Trigonometry Formulas
Trigonometry Terms and Definitions

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

Explore Trigonometry formulas with explanations and examples

Sine Function (sin)

sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

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

Secant Function (sec)

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

Cotangent Function (cot)

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

Secant Reciprocal Identity

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

Cosecant Reciprocal Identity

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

Cotangent Reciprocal Identity

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

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

Tangent Pythagorean Identity

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta

Secant Even Identity

sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \theta

Sine Odd Identity

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta

Tangent Odd Identity

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \theta

Sine Function (sin)

sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

Cosine Function (cos)

cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

Tangent Function (tan)

tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

Cosecant Function (csc)

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

Secant Function (sec)

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

Cotangent Function (cot)

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

Secant Reciprocal Identity

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

Cosecant Reciprocal Identity

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

Cotangent Reciprocal Identity

cot⁡θ=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Primary Pythagorean Identity

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

Tangent Pythagorean Identity

1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta

Cotangent Pythagorean Identity

1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

Sine-Cosine Co-Function Identity

sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta

Cosine-Sine Co-Function Identity

cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta

Tangent-Cotangent Co-Function Identity

tan⁡(π2−θ)=cot⁡θ\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta

Cotangent-Tangent Co-Function Identity

cot⁡(π2−θ)=tan⁡θ\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta

Secant-Cosecant Co-Function Identity

sec⁡(π2−θ)=csc⁡θ\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta

Cosecant-Secant Co-Function Identity

csc⁡(π2−θ)=sec⁡θ\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta

Cosine Even Identity

cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta

Secant Even Identity

sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \theta

Sine Odd Identity

sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta

Tangent Odd Identity

tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \theta

Cosecant Odd Identity

csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \theta

Cotangent Odd Identity

cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \theta

Trigonometry Terms and Definitions

Explore Trigonometry formulas with explanations and examples

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.