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. Trigonometric functions are also defined for angles $\theta$ that are not acute by using the appropriate reference angle ${\theta }^{\prime }.$ For input values that are not between $0$ and $2\pi ,$ the value of the coterminal angle that belongs to this interval is used instead. Therefore, trigonometric functions are periodic functions.

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 $y\text{-}$coordinate of the point $P.$ Consider the function $y=a\sin b\theta ,$ where $a$ and $b$ are non-zero real numbers and $\theta$ is measured in radians. With this information, the properties of the sine function can be defined.

Properties of $y=a\sin b\theta$
Amplitude	$|a|$
Number of cycles in $[0,2\pi ]$	$|b|$
Period	$\frac{2\pi }{|b|}$
Domain	All real numbers
Range	$[\text{-}|a|,|a|]$

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 $x\text{-}$coordinate of the point $P.$ Consider the function $y=a\cos b\theta ,$ where $a$ and $b$ are non-zero real numbers and $\theta$ is measured in radians. With this information, the properties of the cosine function can be defined.

Properties of $y=a\cos b\theta$
Amplitude	$|a|$
Number of cycles in $[0,2\pi ]$	$|b|$
Period	$\frac{2\pi }{|b|}$
Domain	All real numbers
Range	$[\text{-}|a|,|a|]$

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 $y\text{-}$coordinate to the $x\text{-}$coordinate of the point $P.$ Consider the function $y=a\tan bx,$ where $a$ and $b$ are non-zero real numbers and $\theta$ is measured in radians. The properties of the tangent function can be identified from the function rule.

Properties of $y=a\tan bx$
Amplitude	No amplitude
Interval of One Cycle	$(\text{-}\frac{\pi }{2|b|},\frac{\pi }{2|b|})$
Asymptotes	At the end of each cycle
Period	$\frac{\pi }{|b|}$
Domain	All real numbers except odd multiples of $\frac{\pi }{2|b|}$
Range	All real numbers

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 $y\text{-}$coordinate of $P.$

Consider the general form of a cosecant function.

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 $y=a\csc bx$
Amplitude	No amplitude
Number of Cycles in $[0,2\pi ]$	$|b|$
Period	$\frac{2\pi }{|b|}$
Domain	All real numbers except multiples of $\frac{\pi }{|b|}$
Range	$(\text{-}\infty ,\text{-}|a|]\cup [|a|,\infty )$

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 $x\text{-}$coordinate of $P.$

Consider the general form of a secant function.

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 $y=a\sec bx$
Amplitude	No amplitude
Number of Cycles in $[0,2\pi ]$	$|b|$
Period	$\frac{2\pi }{|b|}$
Domain	All real numbers except odd multiples of $\frac{\pi }{2|b|}$
Range	$(\text{-}\infty ,\text{-}|a|]\cup [|a|,\infty )$

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 $x\text{-}$coordinate to the $y\text{-}$coordinate of $P.$

Consider now the general form of a cotangent function.

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 $y=a\cot bx$
Amplitude	No amplitude
Number of Cycles in $[0,2\pi ]$	$2|b|$
Period	$\frac{\pi }{|b|}$
Domain	All real numbers except multiples of $\frac{\pi }{|b|}$
Range	All real numbers