Cosecant

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.

Geometric Representation of the Cosecant

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.

Trigonometric Identities Involving the Cosecant Function

Cosecant of Complementary, Supplementary and Conjugate Angles

• Cosecant of a Complementary Angle:
  
• Cosecant of a Supplementary Angle:
  
• Cosecant of a Conjugate Angle:
  

Cosecant of Negative Angles

• Cosecant of a Negative Angle:
  

Cosecant of Angles that Differs by 90º or 180º

• Cosecant of an Angle that Differs by 90º:
  
• Cosecant of an Angle that Differs by 180º:
  

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.