The Polar Form of a Complex Number

Oct 13, 2025 by windailys.com 90 views

Introduction

In mathematics, complex numbers are used to represent points in a two-dimensional plane. They are of the form ${$ a + bi$}$, where ${$ a$}$ and ${$ b$}$ are real numbers, and ${$ i$}$ is the imaginary unit, which satisfies ${$ i^2 = -1$}$. The polar form of a complex number is a way to express it in terms of its magnitude (or length) and angle (or direction) from the origin. In this article, we will determine the polar form of the complex number ${ $5 - 3i\$}$ and express the angle in degrees.

The Polar Form of a Complex Number

The polar form of a complex number ${$ a + bi$}$ is given by:

${$ r(\cos \theta + i \sin \theta)$]

where [ $r\$}$ is the magnitude (or length) of the complex number, and ${$ \theta$}$ is the angle (or direction) from the origin.

Calculating the Magnitude

The magnitude of a complex number ${$ a + bi$}$ is given by:

${$ r = \sqrt{a^2 + b^2}$]

In this case, we have [ $a = 5\$}$ and ${$ b = -3$}$. Plugging these values into the formula, we get:

${$ r = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}$]

Calculating the Angle

The angle [ $\theta\$}$ can be calculated using the formula:

${$ \theta = \tan^{-1} \left(\frac{b}{a}\right)$]

In this case, we have [ $a = 5\$}$ and ${$ b = -3$}$. Plugging these values into the formula, we get:

${$ \theta = \tan^{-1} \left(\frac{-3}{5}\right)$]

Using a calculator, we find that:

[ $\theta \approx -0.588}$ $ radians

To convert this angle to degrees, we multiply by ${$ \frac{180}{\pi}$]:

[$\theta \approx -0.588 \times \frac{180}{\pi} \approx -33.69^{\circ}$]

However, we are asked to express the angle in the range [$0 \leq \theta \leq 360^{\circ}$]. To do this, we add [$360^{\circ}$] to the angle:

[$\theta \approx -33.69^{\circ} + 360^{\circ} \approx 326.31^{\circ}$]

Rounding Numerical Entries

We are asked to round numerical entries in the answer to two decimal places. Therefore, we round the magnitude and angle to two decimal places:

[ $r \approx 5.83}$ $ and ${$ \theta \approx 326.31^{\circ}$]

Conclusion

In this article, we determined the polar form of the complex number [ $5 - 3i\$}$ and expressed the angle in degrees. We calculated the magnitude and angle using the formulas for the polar form of a complex number, and rounded the numerical entries to two decimal places. The polar form of the complex number is:

${$5.83(\cos 326.31^{\circ} + i \sin 326.31^{\circ})$]

Introduction

In our previous article, we determined the polar form of the complex number [ $5 - 3i\$}$ and expressed the angle in degrees. 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 ${$ a + bi$}$ is given by:

${$ r(\cos \theta + i \sin \theta)$]

where [ $r\$}$ is the magnitude (or length) of the complex number, and ${$ \theta$}$ is the angle (or direction) from the origin.

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

A: The magnitude of a complex number ${$ a + bi$}$ is given by:

${$ r = \sqrt{a^2 + b^2}$]

Q: How do I calculate the angle of a complex number?

A: The angle [ $\theta\$}$ can be calculated using the formula:

${$ \theta = \tan^{-1} \left(\frac{b}{a}\right)$]

Q: What if the angle is negative?

A: If the angle is negative, you can add [$360^{\circ}$] to the angle to get the equivalent positive angle in the range [$0 \leq \theta \leq 360^{\circ}$].

Q: How do I round numerical entries in the answer to two decimal places?

A: To round numerical entries in the answer to two decimal places, you can use a calculator or round the values manually.

Q: What is the polar form of the complex number [ $-3 + 4i\$}$ ?

A: To determine the polar form of the complex number ${$ -3 + 4i$}$, we can calculate the magnitude and angle using the formulas:

${$ r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$]

[ $\theta = \tan^{-1} \left(\frac{4}{-3}\right) \approx -0.927}$ $ radians

To convert this angle to degrees, we multiply by ${$ \frac{180}{\pi}$]:

[$\theta \approx -0.927 \times \frac{180}{\pi} \approx -53.13^{\circ}$]

However, we are asked to express the angle in the range [$0 \leq \theta \leq 360^{\circ}$]. To do this, we add [$360^{\circ}$] to the angle:

[$\theta \approx -53.13^{\circ} + 360^{\circ} \approx 306.87^{\circ}$]

Rounding the magnitude and angle to two decimal places, we get:

[ $r \approx 5.00}$ $ and ${$ \theta \approx 306.87^{\circ}$]

Therefore, the polar form of the complex number [ $-3 + 4i\$}$ is:

${$5.00(\cos 306.87^{\circ} + i \sin 306.87^{\circ})$]

Q: What is the polar form of the complex number [ $2 - 2i\$}$ ?

A: To determine the polar form of the complex number ${ $2 - 2i\$}$ , we can calculate the magnitude and angle using the formulas:

${$ r = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83$]

[$\theta = \tan^{-1} \left(\frac{-2}{2}\right) = \tan^{-1} (-1) \approx -45^{\circ}$]

However, we are asked to express the angle in the range [$0 \leq \theta \leq 360^{\circ}$]. To do this, we add [$360^{\circ}$] to the angle:

[$\theta \approx -45^{\circ} + 360^{\circ} \approx 315^{\circ}$]

Rounding the magnitude and angle to two decimal places, we get:

[ $r \approx 2.83}$ $ and ${$ \theta \approx 315.00^{\circ}$]

Therefore, the polar form of the complex number [ $2 - 2i\$}$ is:

[$2.83(\cos 315.00^{\circ} + i \sin 315.00^{\circ})$]

Conclusion

In this article, we answered some frequently asked questions about the polar form of complex numbers. We provided examples of how to calculate the magnitude and angle of a complex number, and how to round numerical entries in the answer to two decimal places. We also provided examples of how to determine the polar form of complex numbers using the formulas.