Sine and Cosine Functions are the fundamental aspects that form the basis for Trigonometry. In a general sense, they relate to the ratio of lengths of a right-angled triangle, i.e. sine of an angle is the ratio of the side opposite to the angle the hypotenuse of the triangle and the cosine of an angle is the ratio of the base and hypotenuse of the triangle.
They are used to represent variations of certain physical quantities such as displacement in simple harmonic motion, alternating current, sound waves, etc. Additionally, sine and cosine functions find applications in computer graphics, signal processing, simulation, encryption and decryption, etc.
In this article, we will briefly discuss various real-life applications of sine and cosine functions.
What is Sine Function?
Sine Function is a trigonometric function which takes an angle as input and tells the ratio of the length of the side opposite to the angle and of the hypotenuse in a right-angled triangle. Sine function can be used to express periodic variation of certain physical quantities as sin(ωt), where ω represents the frequency of the change in physical quantity. The periodic variations which are expressed using sine functions are called sinusoidal variations.
Sine Function
Domain of a sine function can be any real number (angle expressed as radians or degrees). The range of a sine function lies between -1 and 1. Sine function is an odd function which means that the sine of a negative angle is negative of the sine of the positive of that angle, i.e. sin(-x) = -sin(x).
What is Cosine Function?
Cosine Function is a trigonometric function taking an angle as input and giving output as the ratio of the length of the side adjacent to the angle and hypotenuse in a right-angled triangle. A cosine function can also be used to represent periodic variations with specific boundary conditions. Periodic variations represented by cosine functions are also called sinusoidal variations.
Cosine Function
Domain of a cosine function can also be any real number and the range lies between -1 and 1. But, the cosine function is an even function which implies that the cosine of a negative angle is the same as that of a positive angle with the same magnitude, i.e. cos(-x) = cos(x).
Applications of Sine and Cosine Functions in Real-Life
Various application of Sine and Cosine Functions in Real-Life includes:
Healthcare Systems
Sine and Cosine functions are used in various healthcare systems such as medical imaging techniques (MRI, CT scans) to study various electromagnetic signals which are generated by human body and diagnose of any kind of disease in the body. Tissues present in human body generate electromagnetic waves in presence of certain magnetic and electric fields which are represented using sine and cosine functions and analysed in terms of their amplitude, phase, and frequency.
Simple Harmonic Motion
Sine and Cosine Functions can be used to understand the motion of bodies which exhibit a simple harmonic motion. This kind of motion is shown by a variety of bodies in our day to day life. Examples include the motion of a pendulum of a clock, tuning forks, musical instruments, and spring mass systems used in automobile suspension systems, mechanical clocks, shock absorbers, etc. Sine and Cosine functions help to represent the periodicity in motion for bodies executing simple harmonic motion.
Mechanical Engineering
Sine and Cosine Functions find less or more applications in each branch of engineering. In mechanical engineering, we deal with motions of different bodies, i.e. kinematics and dynamics of the motion. To represent rotary motions of shafts, gears, cams, and crankshafts, we use sine and cosine functions. Also, the functions are used to model mechanical vibrations which are further used to manufacture vibration isolators and dampers. In automated manufacturing systems which use conveyors, robotic arms, and material handlers, the functions are used to analyze the behaviour of such systems, e.g. controlling the periodic tasks by representing them in form of sine and cosine functions.
Sound Waves
Sound Waves are a type of mechanical wave which propagate through a medium such as air, water, and solids. These are expressed graphically as waveforms which describe the variation of displacement of particles through which sound travels with respect to time. The waveforms are of periodic type, and thus can be represented through sine and cosine functions. This representation is then used in sound synthesis where complex sounds are generated for use in music production, audio engineering, and digital signal processing.
Computer Graphics
In the field of computer graphics, sine and cosine functions are extensively used to create high definition animations, lightning effects, and simulate various natural phenomena such as ripples, formation of clouds, etc. In engineering graphics, we need to build a 3D model of physical bodies which require construction of specific curves and splines. These are defined using sine and cosine functions in backend code of the software in terms of parametric curves.
Signal Processing
Sine and Cosine functions are also used to represent a variety of signals. The functions can be used to analyze and manipulate different types of signals. For instance, to break a complex signal into smaller sinusodial components, Fourier series is an important tool. To remove unwanted components of a signal, filters are designed with help of sine and cosine functions. Further, modulation and demodulation also relies on sine and cosine functions as frequency, phase, and amplitude can be represented by the function for a signal.
FAQs – Applications of Sine and Cosine Functions
What are applications of sine and cosine functions?
Sine and cosine functions are used to represent periodic variations as these functions are periodic in nature, that their output repeats after a certain interval of input. Periodic phenomena such as light waves, sound waves, simple harmonic motions, alternating currents, etc. can be represented using sine and cosine functions.
Where do sine and cosine functions find applications in mechanical engineering?
In mechanical engineering, there are various systems which undergo periodic motion such as pendulums, shafts, engines, etc. Sine and cosine functions are used to analyze such motions and design them as per specifications.
How do sine and cosine functions used in electrical engineering?
In electrical engineering, we have different types of signal which have a periodic kind of nature, for instance alternating current (AC) signals can be represented using sine and cosine functions. Various circuits involving AC signal can thus be designed and analyzed with help of sine and cosine functions.
Do sine and cosine functions find some application in astronomy?
Yes, sine and cosine functions can be used to represent planetary motions, ocean tides, satellite motions, etc. which are used in the field of astronomy to understand various phenomena related to celestial bodies.
Are sine and cosine functions used in data analysis?
Yes, sine and cosine functions are used in data analysis in form of time series analysis, harmonic analysis, frequency domain analysis, and signal processing. The functions are particularly useful to represent periodic and cyclic datasets.