Sine

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Drawing of the right triangle to calculate the sine

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

Sine 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 sin (from the Latin sinus).

Sines for Special Common Angles

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

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

Properties of Sine

  • Domain: Sine domain (all real numbers).
  • Range: Sine range
  • Symmetry: since sin (-x) = -sin (x) then sin (x) is an odd function and its graph is symmetric with respect to the origin (0, 0).
  • Increasing-decreasing behaviour: over one period and from 0 to 2π, sin (x) is increasing on the intervals (0, π/2) and (3π/2, 2π), and decreasing on the interval (π/2, 3π/2).
  • 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 sine function: The derivative of sine function
  • The integral of sine function: The integral of sine function

Graphical Representation of the Sine Function

Graph of the sine function

The sine 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 Sine

Drawing of the geometric representation of the sine

Relationship Between Sine and Other Trigonometric Functions

There are some basic trigonometric identities involving sine:

(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 Sine Function

Sine of Complementary, Supplementary and Conjugate Angles

Sine of Negative Angles

Sine of Angles that Differs by 90º or 180º

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

Sum to Product and Product to Sum Identities for Sine

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

Law of Sines

The Law of Sines (or Sine Rule) relates the sides of a triangle with the sine of their opposite angles. It states that:

In a triangle ABC, each side (a, b, and c) is directly proportional to the sine of its opposite angle (A, B, and C).

Drawing of the equilateral triangle with its sides and interior angles

Law of Sines formula

Drawing of the triangle circumscribed in a circle

The law of sines ratio is also equal to the diameter (twice the radius, 2R) of the circumference (L) in which the triangle is circumscribed.

That is, all the ratios between each side (a, b, and c) and the sine of the opposite angle (A, B, and C) are directly proportional and this proportion is 2R. This is sometimes written as:

Formula of the sine theorem being the ratios proportional to the diameter of the circumference in which the triangle is circumscribed

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 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 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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