Sine Function

In trigonometry, the sine function is one of three primary functions, the others being cosine and tan. The ratio of the opposite side of a right triangle to its hypotenuse is known as the sine x or sine theta. 

Sine function:

In a right-angled triangle, the sine of an angle is the ratio of the hypotenuse to the side opposite the angle. The sine function has a period of 2π and is an essential periodic function in trigonometry. It gives the value of the sine function of the right triangle’s angle between the base and hypotenuse. It is denoted as sin x in mathematics, where x is the acute angle formed by the base and hypotenuse of a right triangle. 

The sine function for any right triangle with an angle, say ABC, is: 

Sin a = Opposite/Hypotenuse

Sine function formula:

The sine of an angle in a right-angled triangle is equal to the ratio of the side opposite the angle (also known as perpendicular) to the hypotenuse. 

Assume that in a right triangle ABC, ‘x’ is the angle. 

As the sine formula is given by:

Sin a = Opposite side/Hypotenuse

Or

Sin a = Perpendicular/Hypotenuse

The sine formula becomes, as seen in the diagram. 

Sin x = O/H

Where ‘O’ is the side of the angle opposite ‘x’  and ‘H’ is the hypotenuse. 

Sine function values table:

Properties of sine function as per quadrants:

Depending on the quadrants, the sine function yields positive or negative values. Sine 270 is negative, while sine 90 is positive, as shown in the table above. The value of the sine function is determined by the quadrants and is positive in the first and second quadrants, but negative in the third and fourth quadrants. 

Sine function Domain and Range:

Because the sine function is defined for all real numbers, the domain of y = sin x is R, which is the set of all real numbers. The closed interval [-1, 1] is the range of the sine function. This is equivalent to -1 ≤ y ≤ 1 or -1 ≤ sin x ≤ 1. The range of this function, on the other hand, can be expressed in quadrants. To find the range of the sine function in different quadrants, look at the table below.

Sine function period and amplitude: 

We can see from the above that the sine function values do not change when x grows (or drops) by an integral multiple of 2π. Thus, 

Sin (2nπ+x) = sin x, n∈Z

Sin x = 0, if x = 0, ±π, ±2π, ±3π,……i.e. when x is an integral multiple of π. 

This can also be written as:

Sin x = 0 implies x = nπ, where n is an integer.

As a result, we may claim that the value of sin x repeats every 2π interval. As a result, sin x has a period of 2π. 

Convert the supplied sine function to the general form while sketching a graph to find the various parameters such as amplitude, phase shift, vertical shift, and period. 

The general format is as follows: 

a sin (bx-c) + d

Where, |a| = amplitude

2π/|b| = period

c/b = phase shift

d = vertical shift

Sin(πx/3), for example, is a Sine function with a period of 6. 

This can be expressed as follows: 

y = sin kx

Period of sin kx = 2π/|k|

2π/|k| = 6

|k| = π/3

Hence, y = sin(πx/3)

We can also calculate the sine function with a period of 3. 

Inverse sine function:

From the aforementioned ratios, the sine inverse function is used to calculate the angle of a right-angled triangle. Arcsine, asin, or sin⁻¹ are the inverses of sine. 

Conclusion:

The sine function and sine waves are frequently used to model cyclic or periodic economic and financial data. In such modelling activities, time is the variable. Similarly, macroeconomic factors like unemployment, labour force participation, and commodity prices are seasonal and cyclical in nature.