What is -5 - 5i in Trigonometric Form?

When dealing with complex numbers, it is often helpful to express them in trigonometric form. The trigonometric form of a complex number is a powerful tool used in various applications, such as signal processing, electrical engineering, and quantum mechanics. In this article, we will explore how to convert the complex number -5 - 5i into its trigonometric form and understand the significance of each component in this representation.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. The real part of the complex number is represented by 'a', and the imaginary part is represented by 'bi'.

Converting to Trigonometric Form

The trigonometric form of a complex number is given by:

z r cis θ

z r (cos θ i sin θ)

where r is the modulus of the complex number, and θ is the argument (or angle) of the complex number.

Modulus and Argument

The modulus r of a complex number z a bi is given by:

r √(a2 b2)

For the complex number -5 - 5i, where a -5 and b -5:

r √((-5)2 (-5)2) √(25 25) √50 5√2

The argument θ is the angle that the complex number makes with the positive real axis in the complex plane. It can be found using:

tan θ b/a

For the complex number -5 - 5i:

tan θ -5/-5 1

θ arctan(1)

Since the complex number is in the third quadrant (both a and b are negative), the angle θ must be:

θ π arctan(1) π π/4 5π/4

The Trigonometric Form

Combining the modulus and argument, we can write the complex number -5 - 5i in trigonometric form:

z 5√2 (cos 5π/4 i sin 5π/4)

This can also be written as:

z 5√2 cis 5π/4

Verification Using Euler's Formula

Euler's formula states that eiθ cos θ i sin θ. Thus, we can also express the complex number as:

z 5√2 ei5π/4

Conclusion

By converting the complex number -5 - 5i into its trigonometric form, we have gained a deeper understanding of its geometric representation in the complex plane. This form is particularly useful in various mathematical and engineering applications, where complex numbers are frequently encountered.

For further reading and practice, consider exploring more complex numbers in trigonometric form and their applications in different fields.