A unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) in the coordinate plane.
The unit circle is a powerful tool in trigonometry because it provides a geometric representation of the trigonometric functions:
- Sine and cosine are defined as the coordinates of a point on the unit circle.
- Tangent is the ratio of the sine and cosine.
The unit circleis a circle with a radius of 1 unit, centered at the origin (0, 0) in the coordinate plane.
The trigonometric functionsare defined for all real numbers, not just angles in a triangle.
Sine and Cosine
The sine of an angle \$\theta\$ is the \$y\$-coordinate of the point on the unit circle corresponding to that angle.
The cosine of an angle \$\theta\$ is the \$x\$-coordinate of the point on the unit circle corresponding to that angle.
The sineand cosinefunctions are periodic with a period of \$2\pi\$.
The sineand cosinefunctions are periodic with a period of \$2\pi\$.
Tangent
The tangent of an angle \$\theta\$ is the ratio of the sine and cosine of that angle:
$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$
The tangentfunction is periodic with a period of \$\pi\$.
The tangentfunction is periodic with a period of \$\pi\$.
Cosecant, Secant, and Cotangent
The cosecant of an angle \$\theta\$ is the reciprocal of the sine of that angle:
$$\csc(\theta) = \frac{1}{\sin(\theta)}$$
The secant of an angle \$\theta\$ is the reciprocal of the cosine of that angle:
$$\sec(\theta) = \frac{1}{\cos(\theta)}$$
The cotangent of an angle \$\theta\$ is the reciprocal of the tangent of that angle:
$$\cot(\theta) = \frac{1}{\tan(\theta)}$$
The cosecant, secant, and cotangentfunctions are undefined when their respective reciprocals are zero.
The cosecant, secant, and cotangentfunctions are undefined when their respective reciprocals are zero.
Reference Angles
A reference angle is the acute angle formed by the terminal side of an angle and the x-axis.
The reference angleis always positive and is always between \$0\$ and \$\frac{\pi}{2}\$ radians (or \$0\$ and \$90^\circ\$).
The reference angleis always positive and is always between \$0\$ and \$\frac{\pi}{2}\$ radians (or \$0\$ and \$90^\circ\$).
Trigonometric Functions of Any Angle
The trigonometric functions of any angle are defined using the unit circle.
The trigonometric functionsof any angle are defined using the unit circle.
The trigonometric functionsof any angle are defined using the unit circle.
Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variable for which the trigonometric functions are defined.
The Pythagorean identitiesare derived from the Pythagorean theorem.
The Pythagorean identitiesare derived from the Pythagorean theorem.
Reciprocal Identities
The reciprocal identities are:
$$\csc(\theta) = \frac{1}{\sin(\theta)}$$
$$\sec(\theta) = \frac{1}{\cos(\theta)}$$
$$\cot(\theta) = \frac{1}{\tan(\theta)}$$
The reciprocal identitiesare derived from the definitions of the reciprocal trigonometric functions.
The reciprocal identitiesare derived from the definitions of the reciprocal trigonometric functions.
Quotient Identities
The quotient identities are:
$$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$
$$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$$
The quotient identitiesare derived from the definitions of the tangent and cotangent functions.
The quotient identitiesare derived from the definitions of the tangent and cotangent functions.
Even-Odd Identities
The even-odd identities are:
$$\sin(-\theta) = -\sin(\theta)$$
$$\cos(-\theta) = \cos(\theta)$$
$$\tan(-\theta) = -\tan(\theta)$$
$$\csc(-\theta) = -\csc(\theta)$$
$$\sec(-\theta) = \sec(\theta)$$
$$\cot(-\theta) = -\cot(\theta)$$
The even-odd identitiesare derived from the symmetry of the unit circle.
The even-odd identitiesare derived from the symmetry of the unit circle.
Co-Function Identities
The co-function identities are:
$$\sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)$$
$$\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)$$
$$\tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta)$$
$$\csc\left(\frac{\pi}{2} - \theta\right) = \sec(\theta)$$
$$\sec\left(\frac{\pi}{2} - \theta\right) = \csc(\theta)$$
$$\cot\left(\frac{\pi}{2} - \theta\right) = \tan(\theta)$$
The co-function identitiesare derived from the symmetry of the unit circle.
The co-function identitiesare derived from the symmetry of the unit circle.
Sum and Difference Identities
The sum and difference identities are:
$$\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$$
$$\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)$$
$$\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$$
$$\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$$
$$\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}$$
$$\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}$$
The sum and difference identitiesare derived from the definitions of the trigonometric functions.
The sum and difference identitiesare derived from the definitions of the trigonometric functions.
Double Angle Identities
The double angle identities are:
$$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$$
$$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$
$$\cos(2\theta) = 2\cos^2(\theta) - 1$$
$$\cos(2\theta) = 1 - 2\sin^2(\theta)$$
$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$
The double angle identitiesare derived from the sum identities.
The double angle identitiesare derived from the sum identities.
Half Angle Identities
The half angle identities are:
$$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$$
$$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$$
$$\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$$
$$\tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)}$$
$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)}$$
The half angle identitiesare derived from the double angle identities.
The half angle identitiesare derived from the double angle identities.
Solving Trigonometric Equations
Trigonometric equations are equations that involve trigonometric functions.
To solve trigonometric equations, use the trigonometric identitiesto simplify the equation and then solve for the variable.
To solve trigonometric equations, use the trigonometric identitiesto simplify the equation and then solve for the variable.
Applications of Trigonometric Functions
Trigonometric functions are used to model periodic phenomena such as sound waves, light waves, and tides.
Trigonometric functionsare used to model periodic phenomena such as sound waves, light waves, and tides.
Trigonometric functionsare used to model periodic phenomena such as sound waves, light waves, and tides.
Graphs of Trigonometric Functions
The graphs of trigonometric functions are periodic and have specific characteristics such as amplitude, period, and phase shift.
The graphs of trigonometric functionsare periodic and have specific characteristics such as amplitude, period, and phase shift.
The graphs of trigonometric functionsare periodic and have specific characteristics such as amplitude, period, and phase shift.
Inverse Trigonometric Functions
The inverse trigonometric functions are the inverses of the trigonometric functions and are used to find angles given the values of the trigonometric functions.
The inverse trigonometric functionsare the inverses of the trigonometric functions and are used to find angles given the values of the trigonometric functions.
The inverse trigonometric functionsare the inverses of the trigonometric functions and are used to find angles given the values of the trigonometric functions.
Trigonometric Form of Complex Numbers
The trigonometric form of complex numbers is a way to represent complex numbers using trigonometric functions.
The trigonometric form of complex numbersis a way to represent complex numbers using trigonometric functions.
The trigonometric form of complex numbersis a way to represent complex numbers using trigonometric functions.
Polar Coordinates
Polar coordinates are a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Parametric Equations
Parametric equations are a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Vectors
Vectors are quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Dot Product
The dot product is a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
Cross Product
The cross product is a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
Matrices
Matrices are rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Determinants
The determinant is a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
Inverse of a Matrix
The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
Systems of Linear Equations
A system of linear equations is a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
Conic Sections
Conic sections are the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Sequences and Series
A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
Probability
Probability is the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Statistics
Statistics is the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Limits
A limit is the value that a function approaches as the input approaches a certain value.
A limitis the value that a function approaches as the input approaches a certain value.
A limitis the value that a function approaches as the input approaches a certain value.
Derivatives
A derivative is a measure of how a function changes as its input changes.
A derivativeis a measure of how a function changes as its input changes.
A derivativeis a measure of how a function changes as its input changes.
Integrals
An integral is a measure of the area under a curve.
An integralis a measure of the area under a curve.
An integralis a measure of the area under a curve.
Differential Equations
A differential equation is an equation that relates a function to its derivatives.
A differential equationis an equation that relates a function to its derivatives.
A differential equationis an equation that relates a function to its derivatives.
Parametric Equations
Parametric equations are a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Polar Coordinates
Polar coordinates are a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Vectors
Vectors are quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Dot Product
The dot product is a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
Cross Product
The cross product is a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
Matrices
Matrices are rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Determinants
The determinant is a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
Inverse of a Matrix
The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
Systems of Linear Equations
A system of linear equations is a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
Conic Sections
Conic sections are the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Sequences and Series
A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
Probability
Probability is the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Statistics
Statistics is the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Limits
A limit is the value that a function approaches as the input approaches a certain value.
A limitis the value that a function approaches as the input approaches a certain value.
A limitis the value that a function approaches as the input approaches a certain value.
Derivatives
A derivative is a measure of how a function changes as its input changes.
A derivativeis a measure of how a function changes as its input changes.
A derivativeis a measure of how a function changes as its input changes.
Integrals
An integral is a measure of the area under a curve.
An integralis a measure of the area under a curve.
An integralis a measure of the area under a curve.
Differential Equations
A differential equation is an equation that relates a function to its derivatives.
A differential equationis an equation that relates a function to its derivatives.
A differential equationis an equation that relates a function to its derivatives.
Parametric Equations
Parametric equations are a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Parametric equationsare a way to represent curves in the plane using a parameter.
Polar Coordinates
Polar coordinates are a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Polar coordinatesare a way to represent points in the plane using a distance from the origin and an angle from the positive x-axis.
Vectors
Vectors are quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Vectorsare quantities that have both magnitude and direction.
Dot Product
The dot product is a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
The dot productis a way to multiply two vectors to get a scalar.
Cross Product
The cross product is a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
The cross productis a way to multiply two vectors to get a vector.
Matrices
Matrices are rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Matricesare rectangular arrays of numbers that can be used to represent systems of linear equations, transformations, and more.
Determinants
The determinant is a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
The determinantis a scalar value that can be computed from a square matrix and is used to determine if the matrix is invertible.
Inverse of a Matrix
The inverse of a matrix is a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
The inverse of a matrixis a matrix that, when multiplied by the original matrix, results in the identity matrix.
Systems of Linear Equations
A system of linear equations is a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
A system of linear equationsis a set of equations that can be solved simultaneously to find the values of the variables.
Conic Sections
Conic sections are the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Conic sectionsare the curves formed by the intersection of a plane and a double-napped cone.
Sequences and Series
A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
A sequenceis an ordered list of numbers, and a seriesis the sum of the terms of a sequence.
Probability
Probability is the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Probabilityis the measure of the likelihood of an event occurring.
Statistics
Statistics is the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Statisticsis the study of collecting, analyzing, interpreting, presenting, and organizing data.
Limits
A limit
