This is a very good and important question that must be addressed with explicitly. I am an Engineering student of one the prestigious University in my country. During my first year in school, we did some elementary Mathematics such as Calculus, Trigonometry, Algebra and many other easy Mathematics. When we got to the second year, the Mathematics topics became more advanced than that of our first year became we did some advanced ODE, Calculus and many others. Similar thing happened in my third year when we did some tough Mathematics like Fourier Series, Laplace Transform, Z Transform and many other advanced topics in Mathematics. I noticed that the higher we go, the more difficult the topics in Mathematics is becoming even though I am now in my fourth year 😊.
After experiencing different difficult topics in Mathematics which frustrated my life, I had to meet one of my lecturer just to ask him why are we doing all this difficult Mathematics? I told him we won't be asked to solve ODE or Calculus in the field just as you've stated "when you start work as an engineer all equipment just operate with logic and user manuals, even just a few of the teams demanded to solve equipment problems when damage occurs, because all jobs will given to vendor."
He then smiled and he responded by telling me that Mathematics is very important in Engineering. He said Engineering is mostly based on design and calculations such that without the background of some fundamental Mathematics, you may not be able to solve real life problems. The conversion became more interesting when he told me that most of the Mathematics we are doing now is just the theory aspect but when we get to the Industry or companies or field, we will surely see the real life application of Engineering Mathematics. He later gave an example of Fourier Series when he said that Fourier Series and Transform is an aspect of electrical and electronics Engineering which can be used to test signals. After hearing all he said, I became perplexed not knowing that Mathematics is very essential in the lives of Engineering students.
I am an Engineering student and I want you to know that I am feeling exactly the way you are feeling because Mathematics isn't that easy to understand if you aren't focused or dedicated to it. I think I fell in love with Mathematics with the help of Engineering. Every Engineering student must have some knowledge about Mathematics so that it will assist them in solving real life problems. Have you also wondered why most of the courses we offer in Engineering are mostly calculations? This is because Engineering is all about solving problem by making life easier and comfortable for everyone in the society with the application of science and technology.
In conclusion, most of the Mathematics we do in Engineering is known as Engineering Mathematics just for the sole purpose of solving Engineering problems even though we might not know its value for now but it will surely help us in our field of Application.
Thanks for reading and I hope this helps.
Simply put it would be used, As the laws and equations used to predict what is going to happen or to build something to meet certain requirements are simply math. "Mathematics is the language of physical science and engineering ".
It can be argued that you do not need to concoct the equations yourself in engineering and you certainly do not need advanced mathematics to use then,
however, Math is a tool to solve problems, problems that would occur in all fields of engineering and all science in general. Hence being able to solve these obstacles that would face you in your projects as an engineer would sometimes need some advanced math.
As a high school student who studies this subject and I can say that mechanics heavily relies on math. it is studied in detail math for god sake, Moreover most of its problems would need a good understanding and usage of trigonometry and vectors and I believe I have not came close to all the applications of math in mechanics so here is that.
If you watch channels like 3blue1brown you would see the applications of math in solving problems and finding proofs. Even if you only use geometry to reach conclusions. I heavily recommend watching to prove my point :
Hope that this mess in anyway helps you know why Mathematics is needed in the faculty of Engineering.