# How to use the CSC function

**What is the CSC function?**

The CSC function calculates the cosecant of an angle (radians). The CSC function returns the same value as 1/SIN(number).

**What is the cosecant?**

The cosecant is one of the trigonometric functions closely related to the sine function.

The cosecant function is the reciprocal of the sine function and is defined as the ratio between the the length of hypotenuse and length of the opposite side of a right triangle.

csc(θ) = hypotenuse / opposite

or

defined as the reciprocal of sine function:

csc(θ) = 1 / sin(θ)

The cosecant can be calculated like this:

csc(θ) = 1 / (sin(θ) / cos(θ)) = cos(θ) / sin(θ)

**What is reciprocal?**

The reciprocal of a number is 1 divided by that number. The reciprocal of a number x is denoted as 1/x or x^{-1}.

**What is the angle θ?**

The Greek letter theta (θ) is commonly used to represent an unknown angle in a right triangle.

**What is a right triangle?**

A right triangle is a type of triangle that contains one internal angle measuring 90 degrees or π/2 radians (a right angle).

**What are radians?**

Radians are a unit used to measure angles. An angle of 1 radian has an arc length equal to the circle's radius.

**What is the relationship between the number pi and radians?**

Radians measure angles by the length of the arc they make in a circle rather than degrees. The full circumference of any circle is 2π multiplied by the circle's radius (2πr).

Since the circumference goes all the way around a circle, that means the full circle measures 2π radians. Half a circle would be π radians (half of 2π). A quarter circle is 2π/4 = π/2 radians. An eighth of a circle is 2π/8 = π/4 radians.

Excel has a function that returns the number pi: PI function

**What are the main trigonometric functions, their domain and range?**

Function |
Domain (input) |
Range (output) |

sin(x) | All real numbers | (-1, 1) |

cos(x) | All real numbers | (-1, 1) |

tan(x) | All real numbers except multiples of π/2 | (-∞, ∞) |

sec(x) | All real numbers except multiples of π | (1, ∞) U (-∞, -1) |

csc(x) | All real numbers except integer multiples of π | (-∞, -1) U (1, ∞) |

cot(x) | All real numbers except integer multiples of π | (-∞, ∞) |

**What are the main trigonometric arcfunctions, their domain and range?**

Function |
Domain (input) |
Range (output) |

sin^{-1}(x) |
[-1, 1] | (-π/2, π/2) |

cos^{-1}(x) |
[-1, 1] | (0, π) |

tan^{-1}(x) |
All real numbers | (-π/2, π/2) |

sec^{-1}(x) |
[-1, 1] | (0, π/2) U (π/2, π) |

csc^{-1}(x) |
[-1, 1] | (-π/2, -0) U (0, π/2) |

cot^{-1}(x) |
All real numbers | (0, π) |

**What is the trigonometric domain?**

The domain of a trigonometric function refers to the set of input values it is defined and valid for. The secant, cosecant, and cotangent functions have restricted domains due to their asymptotes.

**What is the trigonometric range?**

The range of a trigonometric function refers to the set of possible output values it returns.

### Excel Function Syntax

CSC(*number*)

### Arguments

number |
Required. An angle specified in radians. |

### Comments

The CSC function returns

- #VALUE error value if number is non-numeric value.
- #NUM! error value if number is equal or larger than 2^27.

Formula in cell C3:

### Functions in 'Math and trigonometry' category

The CSC function function is one of 73 functions in the 'Math and trigonometry' category.

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