# How to use the CSC function

**What is the CSC function?**

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

#### Table of Contents

## 1. Introduction

**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 also 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 is the opposite side?**

The opposite side is the side opposite to the angle being considered. The image above shows a right-angled triangle, it has three internal angles represented by **A**, **B**, and **C**. The opposite side is determined by the chosen angle **A**, **B** or **C**. **A** has the opposite side a, **B** - b, and **C** - c

**What is the adjacent side?**

The adjacent side is the side that is in contact with the angle being considered and the right angle.

**What is the hypotenuse?**

The hypotenuse is the longest side of the right-angled triangle. It is the side opposite to the right angle (90 degrees).

**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.

## 2. Syntax

CSC(*number*)

number |
Required. An angle specified in radians. |

## 3. Example 1

*Calculate the cosecant of **π/4 radians in a right triangle**? *

The argument is:

- number: C18 which contains π/4 radians or 0.785398163397448 radians.

Formula in cell C20:

**Determine the hypotenuse (c) if the opposite side (b) is equal to 1 and the angle (A) is equal to π/4 radians?**

Cosecant A = c/a

Cosecant π/4 = √2 = 1.414213562

√2 = c/1

If the ratio √2 is equal to c/a then √2 = c/1

c = √2*1

c = √2

c = 1.41421356237309

You can also calculate the hypotenuse using Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side.

c^{2} = a^{2} + b^{2
}b = 1

a = 1

c^{2} = 1^{2} + 1^{2}

c^{2} = 2

c = √2

c = 1.41421356237309

## 4. Example 2

*In a right triangle, if one acute angle measures 30 degrees and the adjacent is 10 units long, find the length of the side opposite to that angle?*

Formula in cell C18:

The argument is:

- number: C18 which represents the angle in radians. Cell C18 contains 0.523598775598299 radians which equals to π/6 radians or 30 degrees.

Formula in cell C20:

Cell C20 returns the cosecant ratio which in this example is 2.

The question tells us that the hypotenuse in the triangle is 10 (c) and the cosecant ratio is 2.

number = adjacent / opposite

2 = c/a

2 = 10/a

a = 10 / 2

a = 5

The length of the opposite side (a) is 5.

## 5. Example 3

*Calculate the cosecant ratio if the adjacent side is 3 and the opposite side is 4 in a right triangle?*

The ATAN function enables you to calculate the angle in radians based on the length of the opposite side and the adjacent side.

=ATAN(4/3)

This formula returns 0.927295218001612 radians which represents the angle A for the triangle displayed in the image above. This angle allows us to calculate the cosecant ratio in cell C20.

The argument is:

- number: C18 which represents the angle in radians. Cell C18 contains 0.927295218001612 radians.

Formula in cell C20:

Cell C20 returns the cosecant ratio which in this example is 1.25 We can calculate the hypotenuse based on this equation:

CSC A = c/a

1.25 = c / 4

c = 1,25 * 4

c = 5

You can also calculate the hypotenuse using Pythagoras theorem which states that the squared hypotenuse is equal to the sum of the squared opposite side and the adjacent side.

c^{2} = a^{2} + b^{2
}b = 3

a = 4

c^{2} = 3^{2} + 4^{2}

c^{2} = 9 + 16

c^{2} = 25

c = √25

c = 5

## 6. Why is the function not working?

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.

### Functions in 'Math and trigonometry' category

The CSC function function is one of 62 functions in the 'Math and trigonometry' category.

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