Tangent

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).

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:

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: (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: (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 integral of tangent function:

Graphical Representation 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

Relationship Between Tangent and Other Trigonometric Functions

There are some basic trigonometric identities involving tangent:

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

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

Trigonometric Identities Involving the Tangent Function

Tangent of Complementary, Supplementary and Conjugate Angles

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

Tangent of Negative Angles

• Tangent of a Negative Angle:
  

Tangent of Angles that Differs by 90º or 180º

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

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

• Angle Sum Identity for Tangent:
  
• Angle Difference Identity for Tangent:
  
• Double-Angle Identity for Tangent:
  
• Half-Angle Identity for Tangent:
  
• Triple-Angle Identity for Tangent:
  

Semiperimeter and Half-angle Formula

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):

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:

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:

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:

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

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:
  
• 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:
  

Trigonometric Ratios for Special Common Angles

The 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.

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

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.