Cosine

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Drawing of the right triangle to calculate the sine

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

Cosine 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 cos (from the Latin cosinus).

Cosines for Special Common Angles

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

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

Properties of Cosine

  • Domain: Cosine domain (all real numbers).
  • Range: Cosine range
  • Symmetry: since cos (-x) = cos (x) then cos (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, cos (x) is decreasing on (0, π) and increasing on the interval (π, 2π).
  • End behaviour: The limits as x approaches ±∞ do not exist since the function values oscillate between +1 and −1. This is a periodic function with period 2π.
  • The derivative of cosine function: The derivative of cosine function
  • The integral of cosine function: The integral of cosine function

Graphical Representation of the Cosine Function

Graph of the cosine function

The cosine 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 Cosine

Drawing of the geometric representation of the cosine

Relationship Between Cosine and Other Trigonometric Functions

There are some basic trigonometric identities involving cosine:

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

Cosine of Complementary, Supplementary and Conjugate Angles

Cosine of Negative Angles

Cosine of Angles that Differs by 90º or 180º

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

Sum to Product and Product to Sum Identities for Cosine

These identities assist in transforming a trigonometric expression presented as a product to a sum or vice versa:

Law of Cosines

The law of cosines (or Cosine Rule) relates one side of a triangle to the other two and the angle they form. The theorem states that:

The square of any side (a, b or c) of a triangle is equal to the sum of the squares of the two remaining sides minus twice their product by the cosine of the angle (A, B or C) they form:

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

Cosine theorem formula

The law of cosines is a generalization of the Pythagorean theorem, which holds only for right triangles. In fact, if A is right angle (90º), its cosine is zero and hence, a2 = b2+c2.

If angle A is obtuse, that is >90º, then the cosine is negative.

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 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 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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