Why study complex numbers in trigonometric form? What is the purpose of studying complex numbers in trigonometric form? I have a few books, but they are vague on this subject. I have the impression that it is easier to calculate powers. Would that be one of the perks?

There are many purposes for studying complex numbers in polar form.
Operations
In mathematics, the trigonometric form is often used for multiplying, dividing, raising to the power, but also for root, exponentiation, logarithm...
Some operations can only be performed with the polar form.
Many things in geometry can also be represented via the trigonometric form using complex numbers, which can simplify their representation and thus make it more understandable. This is especially useful in multidimensional space.
Thanks to the trigonometric form of complex numbers, many useful relationships of trigonometric functions, such as those of trigonometric functions and exponential functions or hyperbolic functions, can be used, which makes calculations easier or, in some cases, the calculation of some operations with a complex argument.
The addition theorems and multiplication theorems for trigonometric functions can be derived from this trigonometric form.
The calculation of integrals with complex integral limits for complex-valued functions is partly only made possible by this trigonometric form, since it allows terms to be transformed in such a way that the integral can simply be drawn without having to consider the whole thing with the unit disks.
Rotation
In addition, things such as rotation in $2D$ can be easily described in the trigonometric form or, in a broader sense, with the quaternion or other extensions, rotation in multidimensional space.
Solutions of polynomials
Many polynomials do not have a simple solution that can be represented as root expressions. The trigonometric form of complex numbers can also be used to derive solutions for polynomials in the form of trigonometric operations, which also makes some polynomials non-numerically solvable.
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