Polar Form of the Complex Number -7 6i: A Comprehensive Guide

When you encounter a complex number in the form of -7 6i, it's important to understand how to represent it in its polar form. This involves converting the number from its Cartesian representation (x yi) to its polar form (rcosθ irsinθ) or using the exponential form (reitheta). In this guide, we'll break down the process, explaining each step in detail.

Understanding the Cartesian View

Imagine you're in a grid city like New York or Chicago. You receive a set of directions: “go 7 blocks West and 6 blocks North.” This is a clear Cartesian view of the world. However, in the world of complex numbers, we move into a different perspective: the Polar view.

The Polar View of the World

In the Polar view, the journey is described not in terms of blocks but in terms of distance and direction. For instance, using a drone, you could travel a certain distance in a specific direction, which is more efficient than traveling through blocks. To convert -7 6i to its polar form, we need to find the distance (r) and the direction (θ).

Calculating the Distance (r)

To find the distance or magnitude of the complex number, we use the Pythagorean theorem. The formula is:

r sqrt(x^2 y^2)

For our complex number -7 6i, we substitute x -7 and y 6:

r sqrt((-7)^2 6^2) sqrt(49 36) sqrt(85)

Determining the Direction (θ)

The direction is a bit trickier. It involves finding the angle in the complex plane. This can be done using the arctangent function (tan-1):

θ arctan-1(6 / -7)

However, since the number -7 6i is in the second quadrant (where x is negative and y is positive), we need to add 180 degrees to the result of arctan(6 / -7). In radians, this would be:

θ π - arctan(6 / 7)

Expressing the Complex Number in Polar Form

Now that we have the values for r and θ, we can express the complex number in its polar form:

-7 6i sqrt(85) cos(π - arctan(6 / 7)) i * sqrt(85) sin(π - arctan(6 / 7))

Alternatively, we can use the exponential form:

-7 6i sqrt(85) * eitheta, where θ π - arctan(6 / 7)

Practical Examples and Applications

Understanding polar form is crucial in various fields, including electrical engineering, signal processing, and control systems. For instance, in electrical engineering, polar form helps in analyzing AC circuits. The polar form simplifies the analysis of sinusoidal functions and complex impedances.

In signal processing, the polar form of a complex number is used to analyze signal amplitudes and phases, which is essential for filtering and modulation techniques.

Conclusion

Converting a complex number like -7 6i to its polar form is a fundamental skill that enhances our understanding of complex numbers. By using the Pythagorean theorem and trigonometric functions, we can represent numbers in a more intuitive and efficient manner.

Key Points

Keywords

Keywords: polar form, complex number, Pythagorean theorem, trigonometry