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).

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:

Properties of Sine

• Domain: (all real numbers).
• 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 integral of sine function:

Graphical Representation 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

Relationship Between Sine and Other Trigonometric Functions

There are some basic trigonometric identities involving sine:

• Relationship between sine and cosine:
  
• Relationship between sine and tangent:
  
• Relationship between sine and tangent of a half-angle:
  
• Relationship between sine and cosecant:
  
• Relationship between sine and secant:
  
• Relationship between sine and cotangent:
  

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

Trigonometric Identities Involving the Sine Function

Sine of Complementary, Supplementary and Conjugate Angles

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

Sine of Negative Angles

• Sine of a Negative Angle:
  

Sine of Angles that Differs by 90º or 180º

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

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

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

• Sum to Product Identities for Sine:
  
  
• Product to Sum Identities for Sine:
  
  

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).

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:

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 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 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.