Secant

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Drawing of the right triangle to calculate the secant

The secant is the reciprocal trigonometric ratio of the cosine. It is the reciprocal or multiplicative inverse of the cosine, that is, sec θ · cos θ = 1.

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

Secant formula

Like other trig functions, it is usually abbreviated. So, in a formula, the secant is abbreviated as sec (secant: from Latin secant, “cutting”, from the verb secare).

Secant for Special Common Angles

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

Table of the secant of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)
Drawing on the goniometric circumference of the secant of the most characteristic angles and the sign of the secant in each quadrant

Properties of Secant

  • Domain: Secant domain (all real numbers), except π/2 + n · π, where n is an integer. Or this casuistry: x ≠ ±π/2; ±3π/2; ±5π/2;… (that is, odd multiples of π/2).
  • Range: Secant range
  • Symmetry: since sec (-x) = sec (x) then sec (x) is an even function and its graph is symmetric with respect to the Y axis.
  • Increasing-decreasing behaviour: over one period and from 0 to 2π, sec (x) is increasing on the intervals (0, π/2) and (π/2,π), and decreasing on the intervals (π, 3π/2) and (3π/2,2π).
  • End behaviour: The limits as x approaches π/2+ 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 secant function: The derivative of secant function
  • The integral of secant function: The integral of secant function

Graphical Representation of the Secant Function

Graph of the secant function

The secant 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 Secant

Drawing of the geometric representation of the secant

Relationship Between Secant and Other Trigonometric Functions

There are some basic trigonometric identities involving secant:

  • Relationship between secant and sine:
    Formula for the relationship of secant to sine
  • Relationship between secant and cosine:
    Formula for the relationship of secant to cosine
  • Relationship between secant and tangent:
    Formula for the relationship of secant to tangent
  • Relationship between secant and cosecant:
    Formula for the relationship of secant to cosecant
  • Relationship between secant and cotangent:
    Formula for the relationship of secant to cotangent

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

Trigonometric Identities Involving the Secant Function

Secant of Complementary, Supplementary and Conjugate Angles

Secant of Negative Angles

Secant of Angles that Differs by 90º or 180º

Other Reciprocal Trigonometric Ratios

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

  • Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.
    Cosecant formula
  • 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.
    Cotangent formula

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:

Table of inverse trigonometric ratios (cosecant, secant, cotangent) of the most characteristic angles (0º, 30º, 45º, 60º, 90º, 180º and 270º)

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:

Table of the relationship between trigonometric ratios

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

Trigonometric Ratios of Angle θ

Drawing of the right triangle to calculate the sine

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:
    Sine formula
  • 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:
    Cosine formula
  • 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:
    Tangent formula

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.

Drawing of the trigonometric functions of a triangle on a circle of radius 1

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


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