The trigonometry is an area that belongs to the mathematics which is responsible for studying all the different relationships that can actually arise between angles and sides having a triangle. For them, it depends on trigonometric ratios between which we can find the sine .

What is the breast?

The sine is a type of trigonometric function that is present in right triangles and that can be measured or found when the opposite leg of the triangle is divided by its hypotenuse .

  • Definition
  • Breast characteristics
  • What is the breast for
  • How it is calculated
  • Trigonometric relationships
  • Derivative
  • Integral
  • Domain and range
  • Inverse sine function
  • Examples


The word sinus comes from the Latin word ” sinus ” and has several uses, in this case, we will focus on the area of mathematics . It is one of the six different trigonometric functions that exist which are also known as circular functions. So, in the field of trigonometry, the sine can be defined as the ratio that exists between the leg opposite the angle and the hypotenuse .

There is also a term related to the sine, which is known as the law of sines and it establishes a relationship of proportionality or constant relationship between the different quantities that can be measured between the length of each of the sides. of the triangle and the sine of each of the opposite angles. It is also known as the sine theorem .

Breast characteristics

Some of the main characteristics of the breast are mentioned below:

  • His dominion is  R .
  • It has a route of [- 1, 1]
  • It has the ability to cut the X axis at points k π with k∈Z
  • It is odd , in other words, it is symmetric with respect to the origin.
  • It is also characterized by being strictly increasing in terms of the intervals of the form (a, b).
  • It is also decreasing .
  • It has infinite maximums relative to the points of the shape.
  • It is bounded above by 1 and below by -1.

What is the breast for

The sine can be used in several fields, mainly scientific, and is considered a quite important trigonometric function. It is used to perform calculations on the distances that are unknown, to measure angles and to know exactly the distances that have been traveled. In real life, it is widely used when you need to make calculations about heights or also to measure the angles of certain objects.

How it is calculated

The formula used to calculate and find the sine is the following:

Sen B = opposite leg / hypotenuse = b / a

Trigonometric relationships

As for the trigonometric relations of the sine, these can also be related to some trigonometric identities. some of them are the following:

  • The sine is an odd function that is represented as sin (-x) = -sen (x)
  • The sine is a type of periodic function that works with period 2 π


In order to find out the derivative of the sine, it is necessary to use the chain rule, which is a simpler procedure than using limits on the derivative of a composition of functions. This rule is represented by the following formula:

z ‘= sin’ (x 2 – 2) = 2x cos (x 2 – 2)

Then, the derivative formula must be applied:

[sin x] ‘= cos x


The integral of the sine is represented as follows:

∫ sin x dx = – cos x + C

Domain and range

The sine domains are considered infinite, so the sine domain and range are as follows:

  • Domain: (-∞, ∞)
  • Range: (-1, 1)

Inverse sine function

The inverse sine function is denoted as follows: y = sin –1 x. This can be read as “y is the inverse of the sine of x”. This sentence means that y is the angle that the real number has and that its sine value is x.


Some examples of sinus are mentioned below:

  • 270: -1 breast
  • 90: 1 sine
  • From 70: 0.93969
  • From 30: 1/2
  • From 60: √3 / 2
  • Sine of pi: 0

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