Polar Form of Complex Numbers: Understanding Quadrants and Sign Conventions

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SUMMARY

The discussion focuses on the polar form of complex numbers, specifically addressing the calculation of the angle \(\alpha\) using the arctangent function \(tan^{-1}(x/a)\). It is established that the signs of \(x\) and \(a\) do not affect the determination of \(\alpha\). The quadrants are clearly defined as: \(\theta = \alpha\), \(\theta = \pi - \alpha\), \(\theta = -\pi - \alpha\), and \(\theta = -\alpha\). The consensus is that the sign of the inputs can be disregarded when calculating \(\alpha\).

PREREQUISITES
  • Understanding of complex numbers and their polar representation
  • Familiarity with the arctangent function \(tan^{-1}\)
  • Knowledge of trigonometric identities and quadrant definitions
  • Basic skills in mathematical notation and operations
NEXT STEPS
  • Study the properties of complex numbers in polar form
  • Learn about the arctangent function and its applications in different quadrants
  • Explore the concept of angle determination in trigonometry
  • Investigate the implications of sign conventions in complex number calculations
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are working with complex numbers and need clarity on angle calculations and quadrant conventions.

Darth Frodo
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Not homework as such, just need some clarification.

When finding [itex]\alpha[/itex] do you have to take the signs into account when finding tan[itex]^{-1}[/itex] x/a. Does it matter if a or x are negative?

Next question is about quadrants

1: [itex]\theta[/itex] = [itex]\alpha[/itex]

2: [itex]\theta[/itex] = [itex]\pi[/itex] - [itex]\alpha[/itex]

3: [itex]\theta[/itex] = -[itex]\pi[/itex] - [itex]\alpha[/itex]

4: [itex]\theta[/itex] = -[itex]\alpha[/itex]

Based on these (if correct) it seems that I should disregard the sign when finding [itex]\alpha[/itex].

Is this correct?
 
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Darth Frodo said:
Not homework as such, just need some clarification.

When finding [itex]\alpha[/itex] do you have to take the signs into account when finding tan[itex]^{-1}[/itex] x/a. Does it matter if a or x are negative?

Next question is about quadrants

1: [itex]\theta[/itex] = [itex]\alpha[/itex]

2: [itex]\theta[/itex] = [itex]\pi[/itex] - [itex]\alpha[/itex]

3: [itex]\theta[/itex] = -[itex]\pi[/itex] - [itex]\alpha[/itex]

4: [itex]\theta[/itex] = -[itex]\alpha[/itex]

Based on these (if correct) it seems that I should disregard the sign when finding [itex]\alpha[/itex].

Is this correct?

No, signs aren't taken into account while finding [itex]\alpha[/itex].

I am not sure about the 3 quadrant.
 

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