Sine and Cosine Waves

# Sine and Cosine Waves

Waves are a fundamental concept of physics, mathematics, and engineering and it is nothing but any disturbance caused in a particular medium without the overall or net movement of particles. These waves carry energy with them.

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The waves are usually repetitive and periodic. Waves have mainly five properties. These properties are amplitude, frequency, period, speed, and wavelength. These properties help us in analyzing the behavior of the waves. Let us know more about sine and cosine waves in detail.

## Types of Waves

Waves are of various types having different properties. Some types of waves are given below:

### Transverse Waves

Here, the medium moves in a direction that is at a right angle to the direction of the wave. Eg. water waves, torsion waves, etc.

### Electromagnetic Waves

Here, the medium can be a vacuum. These are produced by various magnetic and electric fields. Hence, due to the changes which are occurring in these waves periodically, they are called electromagnetic waves.

### Matter Waves

It is also called the De-Broglie wave and it shows the relationship between momentum and wavelength.

### Mechanical Waves

These are the waves that require a specific type of medium in order to propagate and thus they are called mechanical waves.

There are various other types of waves aso which are longitudinal waves, torsion waves, surface waves, compression waves, etc. The sine wave is a type of wave that does not come under either transverse or longitudinal waves.

## What are Sine Waves?

These waves are the most famous and common type of wave which is used in the field of mathematics and engineering. It refers to a graphical representation of a function that oscillates above and below 0 in a uniform and repetitive periodic manner.

The sine function is a type of trigonometric function, where mapping is done from the set of numbers consisting of all non-negative or positive real numbers to the interval between [-1,1], i.e., the sine function takes input of any non-negative real number and gives output as a number between -1 and 1.

The sine function and sine waves are used to model periodic phenomena and processes that follow predictable cyclical patterns. The time period of the sine wave is 2 pi or 360 degrees.

Mathematically if we see, the sine function is nothing but the ratio of the perpendicular of a triangle to the hypotenuse of the same triangle. We denote the sine function as sin(x) or sin x where the value of x can be in degrees or in radians.

Here the value of x is variable and can be both positive or negative numbers but the output will only have a value which is between -1 and 1. For example, sin 0 = 0 , sin 90 = 1, sin 45= 0.707, sin 30 = 0.5, sin 60= 0.866 and sin 270= -1. By analyzing these values we can observe that the output of the sine function varies between 1 and -1.

The main property of the sine function is that it comes back to its original value after a difference of 360 degrees. For example, sin 0 is the same as sin 360 i.e. = 0, similarly sin x = sin 360+x = sin 720+x = sin 1080+x, and so on.

## What are Cosine Waves?

These are the waves that are similar to the sine waves with very slight differences. The cosine function can also take any value as an input but the output will vary between -1 and 1 only. The value of cosine peaks at pi radian and crosses the 0 line at 0.5 pi radian. These waves are pretty much analogous to the sine waves and are equally important and used equally in the fields of engineering and mathematics. This function is denoted by cos(x) or cos x.

Mathematically, cos x is the ratio of the base of a right-angled triangle to the hypotenuse of the same right-angled triangle. The value of x is variable and can be any number be it positive or negative. For example, cos 0 = 1 , cos 90 = 0, cos 45= 0.707, cos 30 = 0.866, cos 60= 0.707 and cos 180= -1. By analyzing these values we can observe that the output of the sine function varies between 1 and -1.

Another important property is that the phase difference between the sine wave and cosine wave is 90 degrees i.e. the value of the sine wave is 90 degrees ahead of the cosine wave.

## Difference between Sine and Cosine Waves

The sine function is nothing but the ratio of the perpendicular of a triangle to the hypotenuse of the same triangle. We denote the sine function as sin(x) or sin x where the value of x can be in degrees or in radians.

On the other hand, Mathematically, cos x is the ratio of the base of a right-angled triangle to the hypotenuse of the same right-angled triangle. The value of x is variable and can be any number be it positive or negative. The sine graph peaks at 0.5pi and the cosine graph peaks at pi. The phase difference between the sine and cosine waves is 0.5pi.

## Applications of Sine and Cosine Waves

As we have already discussed, the sine and cosine functions are the most fundamental functions in the entire field of mathematics and physics. Hence, one can easily see their day-to-day usage in the domains of engineering and architecture.

These waves are very commonly used to model different types of radio waves, tides, musical tones, etc. These waves are also used in the field of communication engineering as carrier waves to carry information from the source to the receiver. Another use of the sine function is in modern-day trading to help recognize the patterns and crossovers of the graphs.

Hence, the sine and cosine waves are an essential part in everyday mathematics. These functions have a very important role in the field of engineering and physics and thus they can not be undermined.

## FAQs on Sine and Cosine Waves

### What are waves?

Waves are a fundamental concept of physics, mathematics, and engineering and it is nothing but any disturbance caused in a particular medium without the overall or net movement of particles. These waves carry energy with them. The waves are usually repetitive and periodic in nature.

### What is the value of sin 135?

Value of sin 135 = 0.707.

### Give some applications of Sine and Cosine waves.

These waves are very commonly used to model different types of radio waves, tides, musical tones etc. These waves are also used in the field of communication engineering as carrier wave to carry the information from the source to the receiver

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