Let $f (x) = / arctan x + / mathop{/text{arccot}} x$. Find $f (0) + f (1) + f (/sqrt{2}) + f (/sqrt{3}) $.

Question

Garrett Rivas

Mathematics

20 March, 13:27

Let $f (x) = / arctan x + / mathop{/text{arccot}} x$. Find $f (0) + f (1) + f (/sqrt{2}) + f (/sqrt{3}) $.

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Answers (1)

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Lena Hicks · Answer 1

By direct computation, arctan (0) = 0, arccot (0) = pi/2, arctan (1) = arccot (1) = pi/4.

Let a = arctan (sqrt (2)). Then cot (3*pi/2 - a) = cot (pi/2 - a) = tan (a) = sqrt (2), so arccot (sqrt (2)) = 3*pi/2 - a. Therefore, arctan (sqrt (2)) + arccot (sqrt (2)) = 3*pi/2.

Similarly, arctan (sqrt (3)) + arccot (sqrt (3)) = 3*pi/2.

So the answer is 0 + pi/2 + pi/4 + pi/4 + 3*pi/2 + 3*pi/2 = 4*pi.