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Arctan equations.

  1. Oct 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the following equations in R.

    arctan(1)=arctan(1/2)+arctan(x)
    arctan(1/2)=arctan(1/3)+arctan(x)
    arctan(1/3)=arctan(1/5)+arctan(x)

    conclude that: pi=8arctan(1/5)+4arctan(1/7)+8arctan(1/8)



    3. The attempt at a solution

    For the first three i just took the tan of both sides and i got x=1/3 for the first one and x=1/7 for the second one and for the third one i got x=1/8. For the last one should i find the arctan(1/5) and the others from the first equations and plug them in or is there something else?
     
  2. jcsd
  3. Oct 13, 2012 #2

    Mark44

    Staff: Mentor

    Substitute the values you found in the three equations:
    arctan(1)=arctan(1/2)+arctan(1/3)
    arctan(1/2)=arctan(1/3)+arctan(1/7)
    arctan(1/3)=arctan(1/5)+arctan(1/8)

    So arctan(1/3) = arctan(1) - arctan(1/2)
    arctan(1/7) = arctan(1/2) - arctan(1/3)
    arctan(1/8) = arctan(1/3) - arctan(1/5)

    Now substitute these values into the expression 8arctan(1/5)+4arctan(1/7)+8arctan(1/8) and see if you can show that this expression simplifies to ##\pi##.
     
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