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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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User Guide

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

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User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Introduction to Complex Numbers

Voice

Voice

Home

Representations of Complex Numbers

»  coordinate form a+ib

»  polar form r(cos theta + i sin theta)
abstracted from a+ib
→  r=sqrt(a^2+b^2)
→  cos theta = a/r
→  sin theta = b/r

### Complex Numbers: Equivalent representations

plain and simple summary

nub

plain and simple summary

nub

dummy

A complex number in the form a+ib  is equivalently given in the form r (cos theta + i sin theta)

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Complex number in the form a+ib can be equivalently represented in the form r(cos theta + i sin theta). This representation, has an advantage to understanding the generic form of complex numbers.

Keep tapping on the content to continue learning.
Starting on learning "Equivalent Representation of complex numbers". ;; Complex number in the form a plus i b can be equivalently represented in the form r cos theta plus i sine theta. This representation, has an advantage to understanding the generic form of complex numbers.

Which of the following equals 2?

• 4-2
• 1+1
• 6/3
• all the above

The answer is 'All the above'.

A number can be equivalently represented with numerical expressions of various forms.

Given hypotenuse is c. Which of the following represent the length of side opposite to the angle theta in a right angle triangle?

• c sin theta
• c tan theta cos theta
• c sqrt(1 - cos^2 theta)
• all the above

The answer is 'all the above'.

A constant can be represented as expressions.

It was established that solution to quadratic equation is in the form a+b i. What is an equivalent expression of this?

• sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)
• The option given above is not correct.

The answer is 'sqrt(a^2+b^2) (a/(sqrt(a^2+b^2)) + b/(sqrt(a^2+b^2)) i)'

A number a+bi is equivalently given as
color(darkorange)(sqrt(a^2+b^2)) (color(blue)(a/(sqrt(a^2+b^2)))
quad quad quad + color(green)(b/(sqrt(a^2+b^2))) i)

This form looks similar to color(blue)(cos) and color(green)(sin) of a right angled triangle.

It is equivalently represented as color(darkorange)(r)(color(blue)(cos theta) + i color(green)(sin theta)) where color(darkorange)(r = sqrt(a^2+b^2)) and theta = tan^(-1)(b/a).

sin theta equals ...

• sin (theta + 2n pi) where n=0,1,2,...
• it is not the above

The answer is 'sin (theta + 2n pi)'

A number in the form a+bi is equivalently given as
quad quad = r (cos (theta+ 2n pi)
quad quad quad quad + i sin (theta+ 2n pi))
where r = sqrt(a^2+b^2), theta = tan^(-1)(b/a)

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Progress

Progress

Which of the following equals 2 ?
minus;4;2
4 minus 2
plus;1
1+1
by;6;3
6 by 3
all;above
all the above
The answer is 'All the above'. ;; A number can be equivalently represented with numerical expressions of various forms.
Given hypotenuse is c . Which of the following represent the length of side opposite to the angle theta in a right angle triangle?
sine;
c sine theta
tan;cos;cause
c tan theta cos theta
square;root;minus;1
c square root of 1 minus cos squared theta
all;above
all the above
The answer is 'all the above'. ;; A constant can be represented as expressions.
It was established that solution to quadratic equation is in the form a+b i . What is an equivalent expression of this?
square;root;plus;a;b;
square root of a squared plus b squared ;; multiplied by ;; a by square root ; + ; b by square root ;; into i
option;above;correct;not
The option given above is not correct.
The answer is the expression given with a and b.
A number a plus b i is equivalently given as square root of a squared plus b squared ;; multiplied by ;; a by square root ; + ; b by square root ;; into i ;; This form looks similar to cos and sine of a right angled triangle. ;; It is equivalently represented as r cos theta plus i sine theta ;; where r equals square root of a squared plus b squared and theta = tan inverse (b by a)
sine theta equals?
sine; theta;plus;2
sine theta plus 2 n pi, n is a whole number
not;above
it is not the above
The answer is sine theta plus 2n pi
A complex number in the form a+i b is equivalently given in the form r (cos theta + i sine theta)
A number in the form a+b i, is equivalently given as r cos theta + 2 n pi , plus i sine (theta+ 2 n pi ;; where r = square root a squared plus b squared ;; , theta = tan inverse b by a

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