# Why is cosine so important in Electrical Engineering?

#### cikalekli

Joined Dec 10, 2020
103
Actually I have been thinking this question since 1 week inside my head.

As far as I know that those functions are coming from "Euler's Equation" but I still concerned that what such kind of real life things are making cos and sin functions are so important for us?

I just know that those functions are used in FM broadcast, television broadcast, WiFi network etc. areas.

But what are the more things like "Fourier analysis" tho.? I haven't learned it in my faculty but I just want to learn the main importance of cos and sin functions for us...

Thanks... ^^

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#### BobTPH

Joined Jun 5, 2013
3,658
Are you asking about cosine as opposed to sine? Because, ordinarily we use sine to talk about a periodic waveform of a single frequency. Cosine is just the sine shifted by 1/4 of a period.

if you are asking about the significance of sine and cosine, they are mathematical functions that represent a repeating waveform of a single frequency, and fourier series allow you to represent any arbitrary repeating function as a sum of sin and cos functions.

Bob

#### cikalekli

Joined Dec 10, 2020
103
Are you asking about cosine as opposed to sine? Because, ordinarily we use sine to talk about a periodic waveform of a single frequency. Cosine is just the sine shifted by 1/4 of a period.

if you are asking about the significance of sine and cosine, they are mathematical functions that represent a repeating waveform of a single frequency, and fourier series allow you to represent any arbitrary repeating function as a sum of sin and cos functions.

Bob
Actually, I tried to ask the significance of both of them, that's right.

Thank you for the explanation.

#### cikalekli

Joined Dec 10, 2020
103
Are you asking about cosine as opposed to sine? Because, ordinarily we use sine to talk about a periodic waveform of a single frequency. Cosine is just the sine shifted by 1/4 of a period.

if you are asking about the significance of sine and cosine, they are mathematical functions that represent a repeating waveform of a single frequency, and fourier series allow you to represent any arbitrary repeating function as a sum of sin and cos functions.

Bob
What is the connection between Euler's equation and Fourier series... I mean do they have any relations each other? I came across with that website:

http://www.songho.ca/math/euler/euler.html

It says that " The complex exponential forms are frequently used in electrical engineering and physics. For example, a periodic signal can be represented as the sum of sine and cosine functions in Fourier analysis, and the movement of a mass attached to a string is also sinusodial. These sinusodial functions can be replaced with complex exponential forms for simpler computation. "

So does it means that the Fourier analysis is coming from also Euler's equation?
(I am just sorry, I haven't seen the Fourier analysis yet. I'll take it in the further term)

#### kennybobby

Joined Mar 22, 2019
75
i don't know if this is the sort of thing you are looking for, but here are examples of how these sine and cosine show up in real life.

From studying Physics we find that everything is in motion, from tiny atomic electrons to the largest planets, everything is moving or vibrating about a position of equilibrium or stability towards a state of lowest potential energy. As it turns out the sine and cosine functions are useful to describe the postion, velocity, and accelerations of such oscillatory motion.

For example the earth spins on its axis once every 24 hrs; one revolution is 360 degrees or 2pi radians. So the relative sunlight and darkness of a point on earth looks something like the cosine function when starting the spin angle at high noon. But the temperature rise during the day lags due to the thermal inertia of warming air. This time lag can be represented by a phase shift of the cosine function, and the sine function has an exactly 90 degree phase shift. So the temperature rise looks something like the Sine of the spin angle.

The tides in the ocean are related to the position of the moon, but there is a time lag due to friction and the inertia of the ocean water. The tides look something like the sine function when starting the spin angle at high moon.

So it turns out that the general solution for almost everything can be formed from a combination of [Cos + Sine] as a mathematical tool for modelling and prediction of natural events and motion. This is the Fourier Series.

Look at the voltage and current response of capacitors and inductors in an AC circuit to see phase shift in electronics and how Sine and Cos can be used to describe and model these components.

As Bob said, fourier series allow you to represent any arbitrary repeating function as a sum of sin and cos functions.

Fourier Analysis is the opposite process of extracting the main frequencies of a signal that consists of these Sine and Cos sums.

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#### cikalekli

Joined Dec 10, 2020
103
i don't know if this is the sort of thing you are looking for, but here are examples of how these sine and cosine show up in real life.

From studying Physics we find that everything is in motion, from tiny atomic electrons to the largest planets, everything is moving or vibrating about a position of equilibrium or stability towards a state of lowest potential energy. As it turns out the sine and cosine functions are useful to describe the postion, velocity, and accelerations of such oscillatory motion.

For example the earth spins on its axis once every 24 hrs; one revolution is 360 degrees or 2pi radians. So the relative sunlight and darkness of a point on earth looks something like the cosine function when starting the spin angle at high noon. But the temperature rise during the day lags due to the thermal inertia of warming air. This time lag can be represented by a phase shift of the cosine function, and the sine function has an exactly 90 degree phase shift. So the temperature rise looks something like the Sine of the spin angle.

The tides in the ocean are related to the position of the moon, but there is a time lag due to friction and the inertia of the ocean water. The tides look something like the sine function when starting the spin angle at high moon.

So it turns out that the general solution for almost everything can be formed from a combination of [Cos + Sine] as a mathematical tool for modelling and prediction of natural events and motion.

Look at the voltage and current response of capacitors and inductors in an AC circuit to see phase shift in electronics and how Sine and Cos can be used to describe and model these components.
Ah... I've become so glad to see those bunch of real-life examples...

I'm considerably appreciated for it.

#### Papabravo

Joined Feb 24, 2006
16,808
Mechanical and electrical systems are often described by differential equations. The simple mechanical system of a spring, a mass, and a damper is analogous to the simple electrical circuit of a resistor, and inductor and a capacitor. The differential equations that describe these systems both have the same form. since they have the same form they have the same type of solutions. They are both harmonic oscillators - sines and cosines.

#### MrChips

Joined Oct 2, 2009
24,199
Actually, it is more basic than that and it relates to the math in every field, not just electrical engineering.
Yes, it goes back to Euler's Equation.

Mathematically, any object may have properties f(a, b, c, d...) that can be spacial and non-spacial.
We are more familiar with spacial coordinates (x, y, z) in three dimensions using Cartesian coordinates.
These can be transformed to polar coordinates (r, θ, Φ).

In nature, every object has some kind of "impedance", not just electrical impedance. There is also mechanical impedance in mechanical systems which can be expressed in the general Euler form:

$$z = re^{iθ}$$

When written in Cartesian coordinates, this becomes:

$$z = r(cosθ + isinθ)$$

As another example, planet Earth has impedance. The global temperature of the planet can be modeled in a system with complex delay mechanisms. Hence we can simulate the temperature using complex math that includes sines and cosines. Real world systems have impedance which is resistance to change. Impedance is the push back that wants to resist change. At steady-state there is no change. That is the definition of steady-state. Once you introduce a perturbation, i.e. change, then you set up dynamics which can be modeled with sines and cosines. This leads to the concept of Fourier Series and Fourier analysis which is another important tool in analyzing dynamic systems.

Another common occurrence of sines and cosines would be wave action on beaches and oceans.

So yes, sines and cosines appear everywhere in real life.

#### MrChips

Joined Oct 2, 2009
24,199
@cikalekli
You cannot delete a thread once it is posted.
Deletion of a post or entire thread is at the discretion of the AAC moderation team.
Your question and ensuing responses might be helpful to others having the same questions.

#### drc_567

Joined Dec 29, 2008
1,154
... A different, but frequent use of the cosine function is to determine the 'power factor' of an electrical circuit. The circuit will consist of reactive impedance components, like inductors and capacitors, and real power components such as resistors. The cosine of the included angle between the apparent power and the real power is an important quality characteristic of the circuit, 1.0 being optimal and 0.0 being minimal. ( In the figure shown here, the Apparent Power leg, with units of VoltAmps, is that which is actually applied to the circuit input leads, as with the wall plug, generator, or another type of power source.)

#### cikalekli

Joined Dec 10, 2020
103
@cikalekli
You cannot delete a thread once it is posted.
Deletion of a post or entire thread is at the discretion of the AAC moderation team.
Your question and ensuing responses might be helpful to others having the same questions.
I had planned to delete it because I thought the problem I was asking was ridiculous. I am sorry for this.

Also I'm really appreciated with your detailed explanation.

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#### Delta Prime

Joined Nov 15, 2019
1,149
Hello there Being one of the most important equations in mathematics, Euler’s formula certainly has its fair share of interesting applications in different topics. These include, among others:
The famous Euler’s identity.

The exponential form of complex numbers.

Alternate definitions of trigonometric and hyperbolic functions.

Generalization of exponential and logarithmic functions to complex numbers.

Alternate proofs of de Moivre’s theorem and trigonometric additive identities

Beautiful! isn't it.

#### ApacheKid

Joined Jan 12, 2015
418
Actually I have been thinking this question since 1 week inside my head.

As far as I know that those functions are coming from "Euler's Equation" but I still concerned that what such kind of real life things are making cos and sin functions are so important for us?

I just know that those functions are used in FM broadcast, television broadcast, WiFi network etc. areas.

But what are the more things like "Fourier analysis" tho.? I haven't learned it in my faculty but I just want to learn the main importance of cos and sin functions for us...

Thanks... ^^
An interesting answer I found in The Art of Electronics, is that they are important in linear circuits. A linear circuit excited by a sinewave always responds with a sinewave, no other periodic function has that property in a linear circuit.

#### wayneh

Joined Sep 9, 2010
17,153
Scientists see beauty in the world when complex phenomena can be described by simple math equations. Think how amazing it is that every moving object you see in a lifetime is obeying one simple formula, F = m•A and its mathematical permutations.

When the right math doesn't exist, a clever human comes along and invents it. It's tempting to say humans don't invent the math but reveal the underlying math that was there all along. It's an ongoing philosophical debate that may never be resolved.

Anyway, anything that oscillates and has a repeating pattern over time is tough to describe without a clever piece of mathematics, the sine. It could be done but the beauty we crave would be lacking.

#### nsaspook

Joined Aug 27, 2009
8,671
Quadrature (differ in phase by 90 degrees, cos and sin) functions in RF signals that form the basis of
Digital Modulation Concepts

Quadrature signals, also called IQ signals, IQ data or IQ samples, are often used in RF applications. They form the basis of complex RF signal modulation and demodulation, both in hardware and in software, as well as in complex signal analysis. This post looks at the concept of IQ signals and how they are used.

A pair of periodic signals are said to be in “quadrature” when they differ in phase by 90 degrees. The “in-phase” or reference signal is referred to as “I,” and the signal that is shifted by 90 degrees (the signal in quadrature) is called “Q.” What does this mean and why do we care? Let’s break it down by starting with some basics.
Basics of IQ Signals and IQ modulation & demodulation
IQ Signals Part II

#### Delta Prime

Joined Nov 15, 2019
1,149
When the right math doesn't exist, a clever human comes along and invents it.
With respect and humility that is an oxymoron but then again I quoted you piecemeal without the proper context so it's void never mind that was humor a double negative oxymoron I believe I just invented that!