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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 $θ$ 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.

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 | $∣b∣2π $ | |

Domain | All real numbers | |

Range | $[-∣a∣,∣a∣]$ |

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 $x-$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 | $∣b∣2π $ | |

Domain | All real numbers | |

Range | $[-∣a∣,∣a∣]$ |

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 $y-$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. With this information, the properties of the tangent function can be stated.

Properties of $y=atanbx$ | |
---|---|

Amplitude | No amplitude |

Interval of One Cycle | $(-2∣b∣π ,2∣b∣π )$ |

Asymptotes | At the end of each cycle |

Period | $∣b∣π $ |

Domain | All real numbers except odd multiples of $2∣b∣π $ |

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 $y-$coordinate of $P.$

Consider the general form of a cosecant function.

$y=acscbx$

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 | $∣b∣2π $ |

Domain | All real numbers except multiples of $∣b∣π $ |

Range | $(-∞,-∣a∣]∪[∣a∣,∞)$ |

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.

$y=asecbx$

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 $y=asecbx$ | |
---|---|

Amplitude | No amplitude |

Number of Cycles in $[0,2π]$ | $∣b∣$ |

Period | $∣b∣2π $ |

Domain | All real numbers except odd multiples of $2∣b∣π $ |

Range | $(-∞,-∣a∣]∪[∣a∣,∞)$ |

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 | $∣b∣π $ |

Domain | All real numbers except multiples of $∣b∣π $ |

Range | All real numbers |