An Industrial Engineer is an engineer who works in a factory. His job is to solve any problems which come up when the factory is making a product which has been designed by Desing Engineers.
The Design Engineer decides what a product's characteristics should be, builds and tests a prototype (a sample model), and supplies detailed drawings and specifications to Manufacturing (which operates the factory). If the drawings and specs are good enough, Manufacturing will have no problem building hundreds or thousands of products. However, often the specs are incomplete or inconsistent. Then the Industrial Engineer figures out what's wrong with the specs and corrects them so Manufacturing can proceed. Naturally, he usually must confer with the Design Engineers to get the problem solved.
Industrial engineering is a branch of engineering concerned with the development, improvement, implementation and evaluation of integrated systems of people, money, knowledge, information, equipment, energy, material and process. It also deals with designing new prototypes to help save money and make the prototype better. Industrial engineering draws upon the principles and methods of engineering analysis and synthesis, as well as mathematical, physical and Social Sciences together with the principles and methods of engineering analysis and design to specify, predict, and evaluate the results to be obtained from such systems. In lean manufacturing systems, industrial engineers work to eliminate wastes of time, money, materials, energy, and other resources.
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It depends on where in the world, and also who the employer is.
Industrial engineers can estimate the time taken and the process for manufacture. This will help you estimate the type and number of machines required to manufacture your product.
It is important for all engineers to have at least a basic understanding of math. Most very specialized engineers need to have a very good understanding of math in order to do their jobs well and safely.
Engineers use trigonometry anywhere that involves non-right angles. That includes building cars and buildings, bridges, and many other structures. they also use trigonometry when dealing with fields, like magnetic and electric fields. Also, trigonometry is very important with engineers who deal with waves, such as sound or light engineers.
Building engineers use math to help them figure out how big or how small to make the house or whatever they are building.