Tangent

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Drawing of the right triangle to calculate the sine

In a right triangle, the tangent of an angle θ is defined as the ratio of the opposite leg (a) to the adjacent leg (c).

Tangent formula

It is one of the trigonometric ratios. They are called ratios because they are expressed as the quotient of two of the sides of a right triangle.

Its abbreviation is tan (from the Latin tangens, that means “to touch” (Latin verb: tangere)).

Tangents for Special Common Angles

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

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

Properties of Tangent

The tangent is an odd function, unlike the sine and cosine which are even functions. The remaining other functions are odd.

  • Domain: Tangent 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: Cosine range (all real numbers) or (-∞ ,+∞).
  • Symmetry: since tan (-x) = -tan (x) then tan (x) is an odd function and the graph of tan (x) is symmetric with respect to the origin.
  • Increasing-decreasing behaviour: over one period and from 0 to 2π, cos (x) is increasing on (π/2, 3π/2). That is, is creasing between any of the two successive vertical asymptotes. the equations of the vertical asymptotes are x = π/2 + k · π.
  • 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 π.
  • The derivative of tangent function: The derivative of tangent function
  • The integral of tangent function: The integral of tangent function

Graphical Representation of the Tangent Function

Graph of the tangent function

The tangent is a periodic function with period π radians (180 degrees), so this section of the graph will be repeated in different periods.

Geometric Representation of the Tangent

Drawing of the geometric representation of the tangent

Relationship Between Tangent and Other Trigonometric Functions

There are some basic trigonometric identities involving tangent:

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

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

Drawing of the signs of the trigonometric relations in the goniometric circumference

Trigonometric Identities Involving the Tangent Function

Tangent of Complementary, Supplementary and Conjugate Angles

Tangent of Negative Angles

Tangent of Angles that Differs by 90º or 180º

Sum, Difference, Double-Angle, Half-Angle and Triple-Angle Identities for Tangent

Semiperimeter and Half-angle Formula

Drawing of a triangle with its three sides and its three vertices

Using the half-angle formulas we can relate the tangent of the three half angles of a triangle ABC with its three sides (a, b and c):

Briggs formula

This is one of the most important properties of triangles and is, of course, also applicable to the sine and cosine of the half-angle:

Corollary of Briggs formula

A very important corollary of the half-angle formula is Heron’s formula for the area of a triangle.

Law of Tangents

The Law of Tangents is a statement that relates the length of two sides of a triangle with the tangents of the two angles opposite them. It states that:

Drawing of a triangle with its three sides and its three vertices

The ratio between the sum of two sides (a, b, or c) of a triangle and their subtraction is equal to the ratio between the tangent of the mean of the two opposite angles to those sides and the tangent of half their difference. It can be expressed as the following equations:

Tangent theorem formula

The law of tangents is not as known as the law of sines and the law of cosines but is useful to solve for the measures of missing parts of a triangle when you have two sides of a triangle and the angle between them, that is SAS Triangles.

Other Trigonometric Ratios

Drawing of the right triangle to calculate the sine

The trigonometric ratios of an angle θ are the ratios obtained between the three sides of a right triangle. That is the comparison by the quotient of its three sides a, b and c.

  • 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

Trigonometric Ratios for Special Common Angles

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

Table of 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.

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
  • Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1 or sec θ = 1/cos θ.
    Secant 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

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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