The problem involves evaluating the definite integral of the function dx/(x^2+1) over the interval [tan(3), tan(6)]. Participants are exploring the implications of using trigonometric functions and their inverses in this context.
The discussion is active, with participants providing insights into the potential issues with the original calculations. Some guidance has been offered regarding the use of arctan and the importance of specifying the units (degrees or radians) for the angles involved.
There is confusion regarding the interpretation of the angles 3 and 6, with participants noting that tan(3) is greater than tan(6), which affects the sign of the integral's result. The problem's setup may not have clearly defined the units for the angles.
╔(σ_σ)╝ said:arctan( tan(x)) = x
Use this.
Your calculator may be in radians or something.
Ahlahn said:I switched it to degrees and it worked. Wow. Thanks! What's wrong with radians?!
Dick said:Noo. The answer IS approximately -.1415. What's wrong is that arctan(tan(x))=x. That's only true if x is in (-pi/2,pi/2) which is the range of arctan. Otherwise arctan(tan(x)) will give you the number 'a' in (-pi/2,pi/2) such that tan(x)=tan(a). It will differ from x by a multiple of pi. -.1415 may be coming up wrong because it's not the exact answer. You can find an exact answer. Your 'interval' is also a little funny. tan(3)>tan(6). That's why your answer is negative.