Proving a+b+c=abc with Inverse Trig Functions

[ebola_virus]

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.

 

[siddharth]

You should first find what
$$\tan^{-1} x + \tan^{-1} y$$ is.That is, can you find the (??) in the equation below?
$$\tan^{-1} x + \tan^{-1} y = \tan^{-1} (??)$$
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

 

[ebola_virus]

where did x and y come from agn?

 

[siddharth]

x and y are just two variables. You can replace them with a and b if you wish.

 

[ComputerGeek]

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.

so you mean:
$$arctan(a) + arctan(b) + arctan(c) = \pi$$

 

[Tide]

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 $a + b + c = a \cdot b \cdot c$ because the maximum possible value of $a \cdot b \cdot c$ is $(\pi/3)^3$ which is less than $\pi$.

 

[NateTG]

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 $a + b + c = a \cdot b \cdot c$ because the maximum possible value of $a \cdot b \cdot c$ is $(\pi/3)^3$ which is less than $\pi$.

An example solution is:
$$a=b=c=\sqrt{3}$$
so
[tex]\tan^{-1}(a)=\tan^{-1}(b)=\tan^{-1}(c)=\frac{\pi}{3}[/itex]<br /> Then<br /> $$abc=3 \sqrt{3}$$<br /> and<br /> $$a+b+c= 3 \sqrt{3}$$[/tex]

 

[Tide]

Yikes! What was I thinking!

Thanks for catching that, Nate!

 

[ebola_virus]

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 because 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.