Hello, is there any way to calculate arctan (2) without using calcultor?
And what is that way?
short answer: no
you can always use graphical or numeric methods to find the zeros of the function
f(x) = tan(x) - 2
Then there are Taylor Series, etc.
Your best bet is just to use a calculator (online or mechanical), just as you would to fine log(12) or e^3.1 or any other such animal.
But anyway, if arctan(2) is the solution to a problem, just leave it as-is. That way you have the exact answer, instead of just an approximation.
And, it's clear what the value is. Saying that θ = 1.107 does not give much insight into where it came from.
Yes, there is a way to approximate the value of arctan(2) without using a calculator. One method is to use a mathematical series called the Taylor series expansion.
The Taylor series expansion for arctan(x) is:
arctan(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ...
By substituting x = 2 into this series and summing a sufficient number of terms, we can approximate arctan(2).
To calculate the value, you would follow these steps:
1. Construct the sum of the series by starting with the term x = 2 and subtracting the cube of x divided by 3 (2^3/3). Then add the fifth power of x divided by 5 (2^5/5). Continue alternating between subtracting and adding terms with each power of x, dividing by the corresponding odd integer.
2. Keep adding or subtracting terms until you reach a satisfactory level of precision. The more terms you include, the more accurate your approximation will be.
3. To avoid infinite calculations, you can stop adding terms when the terms become insignificantly small compared to the sum obtained so far. For instance, you can choose to stop when the absolute value of the next term is less than a certain threshold, such as 0.0001.
By applying this method, you can approximate the value of arctan(2) without using a calculator.