Complex Numbers And Quadratic Equations

- It's always better to know how knowledge helps us in real life.

1

In Audio Signal Processing

Sound filters are devices that can make a piano key sound louder or softer, based on its frequency. It is the complex numbers that allow the extra dimension of calculation to enable audio filters to work that way.

2

In AC Circuit Analysis

Complex numbers are also used to calculate voltage, current, or resistance in Alternating Current (AC) circuits.

3

Quantum Mechanics

Complex numbers are of great importance to Quantum Mechanics. A system of complex numbers expresses a wave function in quantum mechanics.

 Complex Number Real Life Application

https://www.youtube.com/watch?v=TIZ6IrtYXns

How are complex numbers used in real life?

 

Complex numbers

Complex numbers are in the form of a+ib which is also called a general form of complex numbers. In a+ib form a is a real part of a complex number, b is the imaginary part of a complex number and i is defined as √(-1). There are many forms of complex numbers. They are,

1. General form (Z = a + ib).
2. Polar form (Z = r(cosθ + isinθ)).
3. Exponential form (Z = reiθ).

General form of a complex number

A complex number has a general form that looks like Z=a+ib where a is the real part and b is the imaginary part and i is called iota which is √(-1). 

Complex Plane Representation

The polar form of a complex number

A complex number has a polar form that looks like Z = r(cosθ + isinθ) where rcosθ is the real part and rsinθ is the imaginary part and i is called iota which is √(-1). 

The Exponential form of a complex number

A complex number has an exponential form that looks like Z=reiθ where eiθ can also be written as cosθ+isinθ which looks like a polar form at the end.

Note: We can also be able to convert general form to polar form, exponential vice versa, also able to convert polar form to general form, exponential form vice versa, and also able to convert exponential to general, polar form vice versa. Now we have to discuss where these complex numbers are useful in real life.

Complex numbers in Real life

Complex numbers are used in many varieties of fields in real life. Below given are the most important uses of complex numbers, and their proper explanation is also explained.

1. Complex number is used in Electronics.
2. Complex number is used in Electromagnetism.
3. Complex number is used to simplify the unknown roots if roots are not real for quadratic equations.
4. Complex numbers are used in computer science engineering.
5. Complex number is used in mechanical and civil engineering.
6. Complex numbers are used in control systems.

• Complex numbers in Electronics

In electronics, we are used to representing the general form of a complex number in the circuit having voltage and current. In Electronics circuit is mainly based on current and voltage. Those two elements are put together as single complex numbers. Z = V+ iI is the complex representation of a circuit having both current and voltage where V is the real axis part and I is the imaginary axis part so that we can able to see the comparison of both V and I by representing as a complex number in electronics. Sometimes in RC circuits or RLC circuits, if we want to combine two elements, say for example resistor and inductor we can write it as R + jXL as well as if in case resistor and capacitor complex number representation would be R + j Xc where XL = jwL and Xc = 1/jwc.

• Complex numbers in Electromagnetism

In electromagnetism, the main elements are the electric field and the magnetic field. Those elements are represented in the form of complex numbers where the real number or real axis is represented by an electric field, the magnetic field is represented as an imaginary part of an imaginary axis.

• Complex numbers to find roots in Quadratic equation

A quadratic equation is used to find the roots of the equation. If the quadratic equation has real roots it is fine. If the quadratic equation cannot able find the real roots then roots are said to be not real roots named as imaginary roots. Imaginary roots can be find by the formula (-b + √(b2 – 4ac))/2a, (-b – √(b2 – 4ac))/2a are two complex imaginary roots if quadratic equation cannot able to find real roots. In this application, we can use complex numbers.

• Complex numbers used in Computer science engineering

In Computer science, data place a major role. The data cannot be seen visually because it is in the form of CSV files. Those CSV file data can be seen by using visual representation methods in computer science. That Visual representation is in the real axis and imaginary axis only so complex numbers are used to represent the data in a visual format for computer science technology. In 2D images also we can use complex numbers. Rotation of a point having real part and imaginary part and translation of a point in a 2D image represents the complex number.

• Complex numbers in Mechanical and Civil Engineering

In mechanical and civil engineering designing is the main for cars and buildings. To do such a thing we have to use 2D designing concepts that mainly depend on complex numbers only. Rotations are also used in drawing for a point is represented by a complex number only.

• Complex numbers in control systems

Control systems conversion of the time-domain system to frequency domain is required that is to be done using Laplace transform. In that poles and zeros of the system are addressed using a complex plane having a real and imaginary axis. This is the reason and uses for the complex number used in the control system.

Real life applications of Trigonometry

 

Trigonometry Used in Navigation

Trigonometry is used to set directions such as the north-south or east-west. It tells you what direction to take with the compass to get in a straight direction. It is used in navigation in order to pinpoint a location. It is also used to find the distance of the shore from a point in the sea. It is also used to see the horizon.

Other Uses of Trigonometry

Following are the other real-life applications of Trigonometry:

• It is used in oceanography in calculating the height of tides in oceans.
• The sine and cosine functions are fundamental to the theory of periodic functions, those that describe sound and light waves.
• Calculus is made up of Trigonometry and Algebra.
• Trigonometry can be used to roof a house, to make the roof inclined ( in the case of single individual bungalows) and the height of the roof in buildings, etc.
• It is used in the naval and aviation industries.
• It is used in cartography (creation of maps).
• Also, trigonometry has its applications in satellite systems.

Real life applications of Trigonometry

 

Trigonometry Can Be Used to Measure the Height of a Building or Mountains

If you know the distance from where you observe the building and the angle of elevation you can easily find the height of the building. Similarly, if you have the value of one side and the angle of depression from the top of the building you can find and another side in the triangle, all you need to know is one side and angle of the triangle.

Have you ever played the game, Mario? When you see him so smoothly glide over the roadblocks. He does not really jump straight along the Y-axis, it is a slightly curved path or a parabolic path that he takes to tackle the obstacles on his way. Trigonometry helps Mario jump over these obstacles. As you know Gaming industry is all about IT and computers and hence Trigonometry is of equal importance for these engineers.

How to improve Vocabulary

 

• 
  
• • 1. 1.Develop a reading habit. Vocabulary building is easiest when you encounter words in context. Seeing words appear in a novel or a newspaper article can be far more helpful than seeing them appear on vocabulary lists. Not only do you gain exposure to unfamiliar words; you also see how they’re used.
    2. 2. Use the dictionary and thesaurus. Online dictionaries and thesauruses are helpful resources if used properly. They can jog your memory about synonyms that would actually be better words in the context of what you’re writing. A full dictionary definition can also educate you about antonyms, root words, and related words, which is another way to learn vocabulary.
    3. 3. Play word games. Classic games like Scrabble and Boggle can function as a fun way to expand your English vocabulary. Crossword puzzles can as well. If you really want to be efficient, follow up rounds of these word games with a little note-taking. Keep a list of the different words you learned while playing the game, and then study that list from time to time.
    4. 4. Use flashcards. A quick way to build a large vocabulary is to study a number of words via flashcards. In today’s digital age, a wide array of smartphone apps make flashcards convenient and easy to organize. Aiming for one new word a day is reasonable. You can always go for more, but it may not be reasonable to assimilate dozens of English words every single day.
    5. 5. Subscribe to “word of the day” feeds. Some web platforms will provide you with a word a day—either on a website, an app, or via email—to help you expand your vocabulary. You can add these words to running word lists.
    6. 6. Use mnemonics. A mnemonic device is a form of word association that helps you remember words’ definitions and proper uses. For instance think of the word obsequious which means “attempting to win favor from influential people by flattery.” Break down that word into components: “obse” is the beginning of “obsessed,” “qui” sounds like the French word for “yes” (oui), and “us” is like the word “us.” So you can think of that big word obsequious as “obsessed with saying yes to us”—which is kind of what it means!
    7. 7. Practice using new words in conversation. It’s possible to amass a huge vocabulary without actually knowing how to use words. This means you have to take it upon yourself to put your personal dictionary into use. If you come across an interesting word in your reading, make a point of using it in conversation. By experimenting in low-stakes situations, you can practice the art of word choice and, with a little bit of trial and error, hone in on the right word for a particular context.

TRICKS TO REMEMBER TRIGONOMETRY TABLE

 

Tyndall effect

 

     Engineering Chemistry - I

                                  Unit -III

                      3.2 COLLOIDS

 

click this below link    

                      https://youtu.be/8Xcpq6e8pBY

Real life application of Trigonometry

 

Real life application of trigonometry

The concepts of trigonometry are applied in various fields to estimate distances. Let us look at some examples

Trigonometry in Astronomy

• Astronomers use trigonometry to calculate how far away from Earth stars and planets are.
• Despite the fact that we are aware of the distances between planets and stars, NASA scientists are still employing this mathematical method to construct and launch rockets and space shuttles today.
• Without knowledge of it, there would not be any satellites in orbit or missions to the moon by humans.

Trigonometry in Flight Engineers

• Trigonometry is used to calculate the speed, direction, and slope in order to determine an airplane’s path from landing to takeoff.
• Trigonometry is used to determine the ideal landing and takeoff angles and speeds, even when the wind is blowing.

For example: A pilot signals to an air traffic controller that she wants to land. She wants to know what angle of fall to take when she is currently at 40,000 feet. Her plane is 100,000 feet from the runway, as the air traffic controller can see on the radar.

We can determine the lengths of the hypotenuse and the opposite by looking at the diagram and knowing that we require θ$�$. Only this formula has both hypotenuse and opposite:

sinθ=sinθ=OppositeHypotenuse$\sin �=\sin �=\frac{��������}{����������}$

sinθ=40,0001000,000$\sin �=\frac{40,000}{1000,000}$

sinθ=0.4$\sin �=0.4$

The pilot must descend with a 23-degree angle.

Learn more about inverse trigonometric functions.

Trigonometry in Navigation

• Trigonometry is used to establish directions like the north, south, east, and west. It also tells you the direction to point the compass in order to travel straight forward.
• To locate a certain location, it is used in navigation. It is also employed to calculate the separation between a location in the sea and the shore. It is employed to view the horizon as well.

Trigonometry in Measuring Heights

• Trigonometry can be used to quickly determine the height of a mountain or a tall building.
• By measuring the distance horizontally from the base and calculating the angle of elevation to the top, the height can be calculated. The perpendicular, base, and hypotenuse of a right-angled triangle will all be represented by this arrangement.

A boy is positioned next to a tree. He asks himself, “How tall is that tree?” as he looks up at it.

Using the tangent function, it can be calculated as follows: tan of angle = ratio of distance to tree height. Suppose the angle is θ$�$ after that

Suppose the distance is 30 meters and the angle is 45 degrees.

Height = 30tan45∘$\frac{30}{\tan {45}^{\circ }}$

tan45∘=1$\tan {45}^{\circ }=1$

Therefore, Height is 30 m