perplexus.info :: Just Math : Find the real number

Find the real number (Posted on 2024-04-25)

Determine the possible real value of n such that:

Arctan(1/3)+Arctan(1/4)+Arctan(1/5)+Arctan(1/n)= Arctan(1)

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Analytical solution

| Comment 3 of 5 |

the sum of arctangent formula makes this a snap:

arctan(a)+arctan(b)=arctan((a+b)/(1-ab))

Rewrite the equation as

Arctan(1/3)+Arctan(1/4)+Arctan(1/5)-Arctan(1)= -Arctan(1/n)

then apply the formula 3 times

Arctan(7/11)+Arctan(1/5)-Arctan(1)= -Arctan(1/n)

Arctan(7/11)+Arctan(-2/3)= -Arctan(1/n)

Arctan(-1/47) = Arctan(-1/n)

n=47

Posted by Jer on 2024-04-26 08:51:44

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