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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 θ that are not acute by using the appropriate reference angle θ'.
For input values that are not between 0 and 2π, 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 P be the point of intersection of the unit circle and terminal side of an angle in standard position. The sine function, denoted as sin, can be defined as the y-coordinate of the point P. Consider the function y=asinbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the sine function can be defined.

Properties of y=asinbθ
Amplitude |a|
Number of cycles in [0,2π] |b|
Period 2π/|b|
Domain All real numbers
Range [- |a|,|a|]
Concept

The Cosine Function

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

Properties of y=acosbθ
Amplitude |a|
Number of cycles in [0,2π] |b|
Period 2π/|b|
Domain All real numbers
Range [- |a|,|a| ]
Concept

The Tangent Function

Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The tangent function, denoted as tan, can be defined as the ratio of the y-coordinate to the x-coordinate of the point P. Consider the function y=atanbx, where a and b are non-zero real numbers and θ is measured in radians. The properties of the tangent function can be identified from the function rule.

Properties of y=atanbx
Amplitude No amplitude
Interval of One Cycle (- π/2|b|,π/2|b|)
Asymptotes At the end of each cycle
Period π/|b|
Domain All real numbers except odd multiples of π/2|b|
Range All real numbers
Concept

Cosecant Function

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

Consider the general form of a cosecant function.


y=acsc bx

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

Properties of y=acscbx
Amplitude No amplitude
Number of Cycles in [0,2π] |b|
Period 2π/|b|
Domain All real numbers except multiples of π|b|
Range (-∞,- |a|] ⋃ [|a|,∞)
Concept

Secant Function

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

Consider the general form of a secant function.


y=asec bx

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

Properties of y=asec bx
Amplitude No amplitude
Number of Cycles in [0,2π] |b|
Period 2π/|b|
Domain All real numbers except odd multiples of π2|b|
Range (-∞,- |a|] ⋃ [|a|,∞)
Concept

Cotangent Function

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

Consider now the general form of a cotangent function.


y=acotbx

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

Properties of y=acotbx
Amplitude No amplitude
Number of Cycles in [0,2π] 2|b|
Period π/|b|
Domain All real numbers except multiples of π|b|
Range All real numbers
Exercises

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