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## Homework Statement

I am currently taking a mathematical methods in physics course. We were given a prerequisite inventory on the first day of class. There are 4 problems that we are assumed to be able to do in our head to 4 or 5 digit accuracy. I am not sure of how to compute these:

1/.9997

(.9997)^.5

sin(.025)

cos(.025)

## Homework Equations

??? Perhaps linear approximation? (I can't recall how to use linear approximation)

## The Attempt at a Solution

For the first one, using a calculator, I get 1.00030009 as an answer. Playing around with this and other combinations of numbers have led me to find a type of method ( I say type of method because I have no idea if it is valid for all numbers) to evaluate an expression of this kind.

.9997 = 100-99.0003

Bring the numerator into the position right of the decimal (i.e. numerator =1 then 1. or numerator = 2 then 2.). Then bring the .0003 from 99.0003 to the left of the decimal to get 1.0003. This also works for 1/.9996 = 1.0004. When the numerator is something other than 1, I think you have to multiply the 3 in .0003 by the numerator. I.e. 2/.9997 = 2.0006 or 2/.9996 = 2.0008.

Using my calculator and playing around has led me to believe there is a pattern that I am not fully describing here. I also see a relationship with the numbers past 5 digit accuracy. But I am not sure of precisely what it is.

For the other problems, I am lost. Does anyone know any tricks here? If possible I would like an explanation for a set of rules I can apply to the situation.

I have seen on the internet various "tricks" for evaluating large numbers. An example would be multiplying a 5 digit number like 34578*11 in your head. Often, these "tricks" do not fully explain what is going on. They usually give a set of conditions that need to be applied in order to use their method.

Sorry for totally butchering all terminology. Thanks for any help.