Sine Function in Trigonometry: Sine Function is one of the six trigonometric ratios which can be defined as the ratio of perpendicular to the hypotenuse in any right angle triangle.

As we know, trigonometry is the branch of mathematics that deals with the relationship between the side and angle of any right-angle triangle. In modern-day, trigonometry is used beyond the right-angle triangle, such as in fields like physics, engineering, computer graphics, and astronomy, and even in everyday applications like GPS navigation and architectural design, so is the Sine Function.

In this article, we will discuss the definition, formula, and various values of the Sine Function, as well as key properties like domain and range, period, and the graph of the Sine Function.

What is Sine Function?

The sine function is one of the primary trigonometric functions others being tan x, cos x, sec x, cot x, and cosec x.

People often confuse the sine function with the sign. It is important to note that these two are two different things one is function [Sine Function] and the other is polarity of any number [Sign] of function. Generally, we define the Sine function for an angle (say x) and denote it as Sin(x) where x can be measured in radians or degrees.

Sine Function Definition

For an acute angle, It can be defined as the ratio of the opposite side of right triangle to the hypotenuse.

Sine Function Formula

If we suppose a right-angled triangle, then for an angle theta we define sin(θ) as:

Sin(θ)= Perpendicular or Opposite Side /Hypotenuse

Domain and Range Of Sine Function

The domain of sine function can be defined between all real numbers as sin x is defined for all x in (-∞, ∞). The range of sin x can be defined as [-1, 1] this can be justified as sin x is a ratio and ratio can never exceed 1. The graph of sine function looks like a wave that oscillates between -1 and 1.

• Domain = All real numbers (−∞, ∞)
• Range = [-1, 1]

Period of Sine Function

Since Sine function is a periodic function, we can define the time period after which the values of Sine function begin to repeat. The period of Sine function is 2π and can be written as:

sin (2nπ + x) = sin x n ∈ integer

For instance, sin(2π) = 0. If you add 2π to the ​x, you get sin(2π + 2π), which is sin(4π). Just like sin(2π), sin(4π) = 0. Every time you add or subtract 2π from our ​x​-value, the solution will be the same.

Reciprocal Of Sine Function

The reciprocal of sine function is called cosecant. We can write it as:

1/sin(x) = cosec(x)

OR

cosec(x) = Hypotenuse/Opposite

The product of the reciprocal of sine and the sine function is always equal to 1 which means:

cosec(x) × sin(x) = 1

From the formula of cosec(x), we can note that cosec(x) will be defined for all values of x except where sin(x) is 0, so we can define domain and range of reciprocal of sin(x) as:

• Domain is all real numbers except nπ, where n is an integer.
• The range of the reciprocal of sine is (-∞, -1] U [1, +∞).

Read More about Reciprocal Identities.

Sine Function Graph

The graphical representation of Sine Function:

We can see that Sine is a periodic function which repeats after 2π. The function starts from 0, attains a maximum value of +1 at π/2 until it reaches a the value of 0 at π. It further attains a minima -1 at 3π/2 until it reaches the value 0 at 2π and starts repeating the cycle again. The graph represents an odd function. This means sin(-x)=-sin(x).

The periodicity of the graph is 2π.

Derivative of Sine Function

The derivative of the sine function, denoted as sin (x), with respect to x is the cosine function, denoted as cos(x). In mathematical notation, this relationship is expressed as:

d/dx(sin (x)) = cos (x)

So, if you have a function y = sin (x), then the derivative dy/dx is equal to cos (x). This is a fundamental trigonometric identity in Calculus.

Integral of Sine Function

The integral of the sine function, denoted as sin (x), with respect to x is the negative cosine function, denoted as – cos (x), plus a constant of integration. In mathematical notation, this relationship is expressed as:

∫ sin (x) dx = – cos (x) + C

Here, C is the constant of integration, and it represents an arbitrary constant that can take any real value. When you differentiate – cos (x) + C with respect to x, you get sin (x), verifying the relationship.

Sine Function Values Table

The value associated with each value of θ is given in the table: