Simplify arctan expression

**Spec** · September 15th 2007, 11:24 AM

Simplify:

$3arctan(2) - arctan(2/11)$

I'd know how to do it if it weren't for the 3 infront of arctan(2).

EDIT: I guess I could do it like this:

$arctan(2) + arctan(2) + arctan(2) - arctan(2/11)$

That seems a bit over the top though. I'm still looking for an easier solution.

Also, I've never encountered three arctan terms before:

$arctan(2) + arctan(3) + arctan(4)$

I'm guessing I need to first simplify $arctan(2) + arctan(3)$ , and then use that to get $(arctan(2) + arctan(3)) + arctan(4)$ by using the $tan(u+v)$ rule again. And of course make sure it's in the range of $\pi/2<arctan(2)+arctan(3)+arctan(4)<3\pi/2$

**Krizalid** · September 15th 2007, 12:19 PM

Call $\alpha=3\arctan2$ & $\beta=\arctan\frac2{11}$

Some identity could be useful for this.

The answer is $\pi$

**Spec** · September 15th 2007, 12:49 PM

I already knew the answer and I've found a way to calculate it, so now I'm just looking for an easier solution than the one I proposed.

**ticbol** · September 15th 2007, 01:19 PM

Originally Posted by Spec

Simplify:

$3arctan(2) - arctan(2/11)$

I'd know how to do it if it weren't for the 3 infront of arctan(2).

I am just curious. Can you please show us how to simplify
arctan(2) -arctan(2/11)
then?

**Spec** · September 15th 2007, 01:58 PM

$tan(arctan(2)-arctan(\frac{2}{11})) = \frac{tan(arctan(2)) - tan(arctan(\frac{2}{11}))}{1 + tan(arctan(2)) * tan(arctan(\frac{2}{11}))} =$
$\frac{2 - \frac{2}{11}}{1 + 2 * \frac{2}{11}} = \frac{20}{15} = \frac{4}{3}$

Therefore,
$arctan(2) - arctan(\frac{2}{11}) = arctan(\frac{4}{3}) + n\pi$

Note that's there's only one n for which the solution is within the defined range.

**ticbol** · September 15th 2007, 02:51 PM

Originally Posted by Spec

$tan(arctan(2)-arctan(\frac{2}{11})) = \frac{tan(arctan(2)) - tan(arctan(\frac{2}{11}))}{1 + tan(arctan(2)) * tan(arctan(\frac{2}{11}))} =$
$\frac{2 - \frac{2}{11}}{1 + 2 * \frac{2}{11}} = \frac{20}{15} = \frac{4}{3}$

Therefore,
$arctan(2) - arctan(\frac{2}{11}) = arctan(\frac{4}{3}) + n\pi$

Note that's there's only one n for which the solution is within the defined range.

Umm, I see.

Then 3arctan(2) -arctan(2/11)
= [arctan(2) +arctan(2)] +[arctan(2) -arctan(2/11)]
===> -4/3 +4/3
= 0

Hence,
3arctan(2) -arctan(2/11)
= arctan(0)
= 0 or pi
But pi is the answer.

You don't like that long solution?

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