# Cosecant

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The cosecant is the reciprocal trigonometric ratio of the sine. It is the reciprocal or multiplicative inverse of the sine, that is, csc θ · sin θ = 1.

In a right triangle, the cosecant of the angle θ is defined as the ratio of the hypotenuse (c) to the opposite leg (a):

Like other trig functions, it is usually abbreviated. So, in a formula, the cosecant is abbreviated as csc or cosec (cosecant: from co ”mutually”, and Latin secant, “cutting”, from the verb secare).

## Cosecant for Special Common Angles

The following table gives the values of cosecant for common angles:

## Properties of Cosecant

• Domain: (all real numbers), except n · π, where n is an integer. Or this casuistry: x ≠ ±π; ±2π; ±3π;… (that is, odd multiples of π).
• Range:
• Symmetry: since csc (-x) = -csc (x) then csc (x) is an odd function and its graph is symmetric with respect to the origin (0, 0).
• Increasing-decreasing behaviour: over one period and from 0 to 2π, sec (x) is increasing on the intervals (π/2, π) and (π, 3π/2), and decreasing on the intervals (0, π/2) and (3π/2, 2π).
• End behaviour: The limits as x approaches π+ k · π do not exist since the function values oscillate between -∞ and +∞. Hence, there are no horizontal asymptotes. This is a periodic function with period 2π.
• The derivative of cosecant function:
• The integral of cosecant function:

## Graphical Representation of the Cosecant Function

The cosecant is a periodic function with period 2π radians (360 degrees), so this section of the graph will be repeated in different periods.

## Relationship Between Cosecant and Other Trigonometric Functions

There are some basic trigonometric identities involving cosecant:

• Relationship between cosecant and sine:
• Relationship between cosecant and cosine:
• Relationship between cosecant and tangent:
• Relationship between cosecant and secant:
• Relationship between cosecant and cotangent:

(1) Note: the sign depends on the quadrant of the original angle.

## Other Reciprocal Trigonometric Ratios

Reciprocal trigonometric ratios are the multiplicative inverses of trigonometric ratios. These are:

• Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1.
• Cotangent (cot): It is the reciprocal ratio of the tangent. Also in this case, cot θ · tan θ = 1 or cot θ = 1/tan θ.

## Reciprocal Trigonometric Ratios for Special Common Angles

The inverse trigonometric ratios for the most common angles (0º, 30º, 45º, 60º, 90º, 180º and 270º) are:

## Relationships Between Trigonometric Functions

Any trigonometric ratio can be expressed in terms of any other. The following table shows the formulas with which each one is expressed as a function of the other:

Note: the sign + or – depends on the quadrant of the original angle.

## Trigonometric Ratios of Angle θ

If θ is one of the acute angles in a right triangle ABC, then:

• The sine is the ratio between the length of the oppisite leg (a) divided by the length of the hypotenuse (c). In a formula, it is written simply as sin:
• The cosine is the ratio between the length of the adjacent leg (b) divided by the length of the hypotenuse (c). In a formula, it is written simply as cos:
• The tangent is the ratio between the length of the opposite leg (a) divided by the length of the adjacent leg (b). In a formula, it is written simply as tan:

## Trigonometric Functions

Trigonometric functions are also called circular functions. The reason is that in a right triangle that has been drawn on the coordinate axis, with the vertex of the angle θ at the center of a circle (O), that is, in standard position, its vertex B can go through all the points of this circle.

Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.

AUTHOR: Bernat Requena Serra

YEAR: 2021