1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Trig proof

  1. Dec 5, 2005 #1
    a teacher wrote on the board

    tan-1(a) + tan-1(b) + tan-1(c) = pi
    [they are inverse trig functions btw, not the recipricol 1/tanx = cotx]

    hence prove that

    a + b + c = abc

    wow. do you have any idea where i can start? thanks. i've been uttered clueless.
     
  2. jcsd
  3. Dec 5, 2005 #2

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    You should first find what
    [tex] \tan^{-1} x + \tan^{-1} y [/tex] is.


    That is, can you find the (??) in the equation below?
    [tex] \tan^{-1} x + \tan^{-1} y = \tan^{-1} (??) [/tex]
    Finding the above is very interesting.
    You have to consider the cases when

    xy<1
    xy>1 and x,y >0
    xy>1 and x,y <0
     
    Last edited: Dec 5, 2005
  4. Dec 5, 2005 #3
    where did x and y come from agn?
     
  5. Dec 5, 2005 #4

    siddharth

    User Avatar
    Homework Helper
    Gold Member

    x and y are just two variables. You can replace them with a and b if you wish.
     
  6. Dec 5, 2005 #5
    so you mean:
    [tex]
    arctan(a) + arctan(b) + arctan(c) = \pi
    [/tex]
     
  7. Dec 5, 2005 #6

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    If the first statement is true then a, b and c can be represented by the interior angles of a triangle. There is no triangle for which [itex]a + b + c = a \cdot b \cdot c[/itex] because the maximum possible value of [itex]a \cdot b \cdot c[/itex] is [itex](\pi/3)^3[/itex] which is less than [itex]\pi[/itex].
     
  8. Dec 5, 2005 #7

    NateTG

    User Avatar
    Science Advisor
    Homework Helper

    An example solution is:
    [tex]a=b=c=\sqrt{3}[/tex]
    so
    [tex]\tan^{-1}(a)=\tan^{-1}(b)=\tan^{-1}(c)=\frac{\pi}{3}[/itex]
    Then
    [tex]abc=3 \sqrt{3}[/tex]
    and
    [tex]a+b+c= 3 \sqrt{3}[/tex]
     
  9. Dec 5, 2005 #8

    Tide

    User Avatar
    Science Advisor
    Homework Helper

    Yikes! What was I thinking!

    Thanks for catching that, Nate!
     
  10. Dec 6, 2005 #9
    er, i guess i got it

    hehe i got it that makes more sort of. systematic sense.

    using the double angle for tan thing

    tan [a + b] = tan (a) + tan (b)
    ---------------
    1 - tan(a)tan(b)

    state that tan (a) = A
    tan (b) = B you'll see why later.

    anyways take the arctan of both sides of the double identity for tan and you get

    a + b = arctan [tan (a) + tab (b) / 1 -tan(a)tan(b)]

    now becuase tan (a) = A
    a = arctan A and vice vresa for B

    hence you end up wiht the arctan identity

    arctan (A) + arctan (B) = arctan [(A+B)/(1-AB)]

    and then you use that for a, b, and c you end up with the simple equation that tan pi = 0, hence

    a + b + c - abc = 0
    hence
    a + b + c = abc

    try that method. just for those of you who awanted a more systematic proof and that made more induction sort of sense. thanks again guys.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Trig proof
  1. Trig Proof (Replies: 1)

  2. Trig Proof (Replies: 5)

  3. Trig proof (Replies: 8)

  4. Trig proof (Replies: 5)

  5. Trig Proof (Replies: 4)

Loading...