Given W=\sqrt{2}\left(\cos \left(\frac{\pi}{4}\right)+i \sin \left(\frac{\pi}{4}\right)\right ] And Z=2\left(\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right ], What Is W − Z W-z W − Z Expressed In Polar

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will explore the polar form of complex numbers, which is a powerful tool for simplifying complex calculations.

What is the Polar Form of a Complex Number?

The polar form of a complex number is a way of expressing a complex number in terms of its magnitude (or length) and argument (or angle). It is denoted by r(cos⁡θ+isin⁡θ)r(\cos \theta + i \sin \theta), where rr is the magnitude and θ\theta is the argument.

Given w=2(cos⁡(π4)+isin⁡(π4))w=\sqrt{2}\left(\cos \left(\frac{\pi}{4}\right)+i \sin \left(\frac{\pi}{4}\right)\right) and z=2(cos⁡(π2)+isin⁡(π2))z=2\left(\cos \left(\frac{\pi}{2}\right)+i \sin \left(\frac{\pi}{2}\right)\right), what is w−zw-z expressed in polar form?

To find w−zw-z, we need to subtract the two complex numbers. We can do this by subtracting their magnitudes and arguments separately.

Subtracting Magnitudes

The magnitude of a complex number is its distance from the origin in the complex plane. To subtract the magnitudes, we simply subtract the magnitudes of the two complex numbers.

∣w∣=2∣z∣=2∣w−z∣=∣w∣−∣z∣=2−2\begin{aligned} |w| &= \sqrt{2} \\ |z| &= 2 \\ |w-z| &= |w| - |z| \\ &= \sqrt{2} - 2 \end{aligned}

Subtracting Arguments

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. To subtract the arguments, we simply subtract the arguments of the two complex numbers.

arg⁡(w)=π4arg⁡(z)=π2arg⁡(w−z)=arg⁡(w)−arg⁡(z)=π4−π2=−π4\begin{aligned} \arg(w) &= \frac{\pi}{4} \\ \arg(z) &= \frac{\pi}{2} \\ \arg(w-z) &= \arg(w) - \arg(z) \\ &= \frac{\pi}{4} - \frac{\pi}{2} \\ &= -\frac{\pi}{4} \end{aligned}

Expressing w−zw-z in Polar Form

Now that we have the magnitude and argument of w−zw-z, we can express it in polar form.

w−z=∣w−z∣(cos⁡arg⁡(w−z)+isin⁡arg⁡(w−z))=(2−2)(cos⁡(−π4)+isin⁡(−π4))\begin{aligned} w-z &= |w-z|(\cos \arg(w-z) + i \sin \arg(w-z)) \\ &= (\sqrt{2} - 2)\left(\cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right)\right) \end{aligned}

Simplifying the Expression

We can simplify the expression by using the trigonometric identities cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta) and sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta).

w−z=(2−2)(cos⁡(π4)−isin⁡(π4))=(2−2)(12−i12)=2−22−i2−22=1−2−i(1−2)\begin{aligned} w-z &= (\sqrt{2} - 2)\left(\cos \left(\frac{\pi}{4}\right) - i \sin \left(\frac{\pi}{4}\right)\right) \\ &= (\sqrt{2} - 2)\left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}\right) \\ &= \frac{\sqrt{2} - 2}{\sqrt{2}} - i \frac{\sqrt{2} - 2}{\sqrt{2}} \\ &= 1 - \sqrt{2} - i(1 - \sqrt{2}) \end{aligned}

Conclusion

In this article, we have explored the polar form of complex numbers and used it to find the difference between two complex numbers. We have shown that the polar form is a powerful tool for simplifying complex calculations and have provided a step-by-step guide on how to use it.

Future Work

In future work, we can explore other applications of the polar form of complex numbers, such as finding the product and quotient of complex numbers, and using it to solve equations involving complex numbers.

References

  • [1] "Complex Numbers" by Math Open Reference
  • [2] "Polar Form of Complex Numbers" by Wolfram MathWorld

Glossary

  • Magnitude: The distance of a complex number from the origin in the complex plane.
  • Argument: The angle a complex number makes with the positive real axis in the complex plane.
  • Polar Form: A way of expressing a complex number in terms of its magnitude and argument.

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Introduction

In our previous article, we explored the polar form of complex numbers and used it to find the difference between two complex numbers. In this article, we will answer some frequently asked questions about the polar form of complex numbers.

Q: What is the polar form of a complex number?

A: The polar form of a complex number is a way of expressing a complex number in terms of its magnitude (or length) and argument (or angle). It is denoted by r(cos⁡θ+isin⁡θ)r(\cos \theta + i \sin \theta), where rr is the magnitude and θ\theta is the argument.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you need to find the magnitude and argument of the complex number. The magnitude is the distance of the complex number from the origin in the complex plane, and the argument is the angle the complex number makes with the positive real axis.

Q: How do I find the magnitude of a complex number?

A: To find the magnitude of a complex number, you can use the formula ∣a+bi∣=a2+b2|a+bi| = \sqrt{a^2 + b^2}, where aa and bb are the real and imaginary parts of the complex number.

Q: How do I find the argument of a complex number?

A: To find the argument of a complex number, you can use the formula arg⁡(a+bi)=tan⁡−1(ba)\arg(a+bi) = \tan^{-1}\left(\frac{b}{a}\right), where aa and bb are the real and imaginary parts of the complex number.

Q: What is the difference between the polar form and the rectangular form of a complex number?

A: The polar form and the rectangular form are two different ways of expressing a complex number. The polar form expresses a complex number in terms of its magnitude and argument, while the rectangular form expresses a complex number in terms of its real and imaginary parts.

Q: When should I use the polar form and when should I use the rectangular form?

A: You should use the polar form when you need to perform operations such as multiplication and division of complex numbers, and when you need to express a complex number in terms of its magnitude and argument. You should use the rectangular form when you need to perform operations such as addition and subtraction of complex numbers, and when you need to express a complex number in terms of its real and imaginary parts.

Q: Can I use the polar form to solve equations involving complex numbers?

A: Yes, you can use the polar form to solve equations involving complex numbers. The polar form is a powerful tool for simplifying complex calculations and can be used to solve equations involving complex numbers.

Q: What are some common applications of the polar form of complex numbers?

A: Some common applications of the polar form of complex numbers include:

  • Finding the product and quotient of complex numbers
  • Solving equations involving complex numbers
  • Expressing complex numbers in terms of their magnitude and argument
  • Performing operations such as multiplication and division of complex numbers

Conclusion

In this article, we have answered some frequently asked questions about the polar form of complex numbers. We have provided a step-by-step guide on how to use the polar form to solve equations involving complex numbers and have highlighted some common applications of the polar form.

Future Work

In future work, we can explore other applications of the polar form of complex numbers and provide more examples of how to use it to solve equations involving complex numbers.

References

  • [1] "Complex Numbers" by Math Open Reference
  • [2] "Polar Form of Complex Numbers" by Wolfram MathWorld

Glossary

  • Magnitude: The distance of a complex number from the origin in the complex plane.
  • Argument: The angle a complex number makes with the positive real axis in the complex plane.
  • Polar Form: A way of expressing a complex number in terms of its magnitude and argument.
  • Rectangular Form: A way of expressing a complex number in terms of its real and imaginary parts.