# Secant

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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):

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:

## Properties of 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:
• 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 integral of secant function:

## Graphical Representation 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.

## Relationship Between Secant and Other Trigonometric Functions

There are some basic trigonometric identities involving secant:

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

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