# Cosecant Formula

## Cosec x Formula

If the length of the hypotenuse is divided by the length of the opposite side, it provides the Cosecant of an angle in a right triangle. One of the trigonometric functions in trigonometry, it is represented as Cosec and occasionally abbreviated as csc.

The Formula for Cosec x is Given as:

Cosec x Formula: Cosec X = Hypotenuse / OppositeSide

### What is the Cosecant Formula in Trigonometric Functions?

There are six trigonometric functions in trigonometry among which sin, cos and tan are the primary functions while the sec, cosec and cot are secondary.

Cosecant is the reciprocal of Sin, and thus Cosec x = 1/SinX

### What is a Cosecant Function in Maths?

There are different angle functions. Sine, cosine, cosecant, secant, tangent, and cotangent functions are the six elementary functions that are used and incorporated in trigonometry.

Csc, sec and cot are basically the reciprocal functions of sine, cosine and tangent functions respectively. Of the six possible trigonometric functions, the cosecant function is specifically the reciprocal of the sine function.

The definition however does not mention how large the triangle is. Nonetheless, that's where the key strength of trigonometry lies: even if we scale the triangle to twice its size, the values of the trigonometric functions (csc x among them) will not change.

### The Sine Function

The sine function is basically a ratio dependent on the opposite side to the hypotenuse. In order to calculate the sine of a 45°angle, we can use the 45°-45°-90° triangle. Note that we cannot have a square root in the denominator, so we multiply both top and bottom by square root of 2 that removes the square root in the denominator. This is referred to as rationalizing the denominator.

### Solved Examples Using Cosec x Formula

Example: Calculate Cosec X if Sin x = 5/9

Solution:

As Cosec X = 1/ Sin X

= 1/5/9

= 9/5

So, Cosec X = 9/5

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FAQs (Frequently Asked Questions)

Q1. What is the Inverse of a Cosecant Function?

Answer: For every trigonometry function like the csc, there is an inverse function that works in reverse. These inverse functions consist of the same name but with 'arc' in front. Thus, the inverse of csc is arccsc etc. When we notice "arccsc A", we conclude it as "the angle whose cosecant is A".

Csc 30 = 2.00 Implies: the cosecant of 30 degrees = 2.000

Arccsc 2.0 = 30 Implies: the angle whose cosecant is 2.0= 30 degrees

Occasionally expressed as acsc or csc-1

Q2. How to Find the Cosecant Where the Angles are Greater Than 90°?

Answer: In a right triangle, the two variable angles (Interior angles of a triangle) are always less than 90°. We can still find the cosecant of any angle, regardless of how large, and also the cosecant of negative angles.

Q3. What is the Use of Trigonometric Functions?

Answer: Trigonometric functions are quite useful and applicable in different areas of our world. Many fields employ these functions to work with the relationships of distances dependent on a right triangle; this is where the word triangulation arise from. Forensic Research, Criminal investigations, and video games are just some areas where trigonometric functions are applicable and useful.

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