How do you find the exact value of #arctan(2)#?

1 Answer
George C.
Mar 15, 2018

This is not a rational number of degrees, nor a rational multiple of #pi# radians.

We can write:

#arctan 2 = pi/2 - sum_(k=0)^oo (-1)^k 1/(2^(2k+1)(2k+1))#

Explanation:

#arctan(2)# is an angle in a right angled triangle with sides #"adjacent" = 1#, #"opposite" = 2# and #"hypotenuse" = sqrt(5)#. It is not a rational multiple of #pi# radians nor a rational number of degrees.

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We can represent it as the sum of an infinite series.

Note that:

#arctan x =sum_(k=0)^oo (-1)^k x^(2k+1)/(2k+1) = x - x^3/3+x^5/5-x^7/7+x^9/9-x^11/11+...#

However, this only converges for #abs(x) <= 1#.

To get a series that does converge, we can use:

#tan (pi/2 - x) = 1/tan x#

So:

#arctan(1/x) = pi/2 - arctan x#

and hence:

#arctan 2 = pi/2 - arctan (1/2)#

#color(white)(arctan 2) = pi/2 - sum_(k=0)^oo (-1)^k 1/(2^(2k+1)(2k+1))#

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