# Sine-squared function

From Calculus

## Definition

This function, denoted , is defined as the composite of the square function and the sine function. Explicitly, it is the map:

For brevity, we write as .

## Key data

Item | Value |
---|---|

Default domain | all real numbers, i.e., all of |

range | , i.e., absolute maximum value: 1, absolute minimum value: 0 |

period | , i.e., |

local maximum value and points of attainment | All local maximum values are equal to 1, and are attained at odd integer multiples of . |

local minimum value and points of attainment | All local minimum values are equal to 0, and are attained at integer multiples of . |

points of inflection (both coordinates) | odd multiples of , with value 1/2 at each point. |

derivative | , i.e., double-angle sine function. |

second derivative | |

derivative | times an expression that is or of , depending on the remainder of mod |

antiderivative | |

mean value over a period | 1/2 |

expression as a sinusoidal function plus a constant function | |

interval description based on increase/decrease and concave up/down | For each integer , the interval from to is subdivided into four pieces: : increasing and concave up : increasing and concave down : decreasing and concave down, : decreasing and concave up |