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Reference

Trigonometric Functions

Concept

Trigonometric Functions

Trigonometric functions are functions that relate an input, which represents an acute angle of a right triangle, to a trigonometric ratio of two of the triangle's side lengths. The angle is usually measured in radians.
Unit Circle Trig Ratios
Trigonometric functions are also defined for angles that are not acute by using the appropriate reference angle
For input values that are not between and the value of the coterminal angle that belongs to this interval is used instead. Therefore, trigonometric functions are periodic functions.

Concept

The Sine Function

Let be the point of intersection of the unit circle and terminal side of an angle in standard position. The sine function, denoted as can be defined as the coordinate of the point Consider the function where and are non-zero real numbers and is measured in radians. With this information, the properties of the sine function can be defined.

Properties of
Amplitude
Number of cycles in
Period
Domain All real numbers
Range

Concept

The Cosine Function

Let be the point of intersection of the unit circle and terminal side of an angle in standard position. The cosine function, denoted as can be defined as the coordinate of the point Consider the function where and are non-zero real numbers and is measured in radians. With this information, the properties of the cosine function can be defined.

Properties of
Amplitude
Number of cycles in
Period
Domain All real numbers
Range

Concept

The Tangent Function

Let be the point of intersection of the unit circle and terminal side of an angle in standard position. The tangent function, denoted as can be defined as the ratio of the coordinate to the coordinate of the point Consider the function where and are non-zero real numbers and is measured in radians. With this information, the properties of the tangent function can be stated.

Properties of
Amplitude No amplitude
Interval of One Cycle
Asymptotes At the end of each cycle
Period
Domain All real numbers except odd multiples of
Range All real numbers

Concept

Cosecant Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cosecant function, denoted as is defined as the reciprocal of the coordinate of

Consider the general form of a cosecant function.

Here, and are non-zero real numbers and is measured in radians. The properties of the cosecant function are stated below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except multiples of
Range

Concept

Secant Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The secant function, denoted as is defined as the reciprocal of the coordinate of

Consider the general form of a secant function.

Here, and are non-zero real numbers and is measured in radians. The properties of the secant function are be stated in the table below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except odd multiples of
Range

Concept

Cotangent Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cotangent function, denoted by is defined as the ratio of the coordinate to the coordinate of

Consider now the general form of a cotangent function.

Here, and are non-zero real numbers and is measured in radians. The properties of the cotangent function are stated below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except multiples of
Range All real numbers