Interesting yet challenging proof

In summary, the equation arctan(1/v) = (π/2) - arctan(v) can be proven using the trigonometry of right triangles.
  • #1
Calixto
16
0
how can I show that ... arctan(1/v) = (π/2) - arctan(v) ?
 
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  • #2
well,try giving and playing with the variables until you get a result.

More of a trial and error method.

Einstienear.
 
  • #3
I know I've been doing that, moving around variables and using the unit circle and right triangles, but I cannot seem to come across a substantial reason why they are equal. Any thoughts?
 
  • #4
Do v and n represent anything in particular?

Because your equation only works when you have certain values for v and n.
 
  • #5
i knew this would confuse someone... that "n" you see is π, or pi... sorry. Does that help?
 
  • #6
if you use the log representation for artan (1/x) and artan (x) so

[tex] artan(x)= (2i)^{-1}(log(1+ix)-log(1-ix)) [/tex] and the same replacing x--> 1/x you

get the accurate result.
 
  • #7
Calixto said:
how can I show that ... arctan(1/v) = (π/2) - arctan(v) ?

Are you allowed to use x = arctan(v) => v = tanx ? :smile:
 
  • #8
I'm going to point significantly (*points significantly*) to my signature. The clue is given by tiny-tim (and that is what they want you to use): make a diagram of a right triangle with x as one of the non-right angles and use v, written as v/1 , as the ratio of the sides that would come from finding the tangent of angle x (label the sides of the triangle appropriately).

Now, in the same triangle, what angle has a tangent of 1/v ? What is the relationship between that angle and angle x ?

(And, with all due respect to mhill, while that relationship is true, the math is probably way beyond what is being done in Calixto's course...)
 
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  • #9
Thanks, I understand it now... But could you explain some more what you wrote about the logs and stuff? Just maybe explain where that comes from so I can impress my teacher :)
 
  • #10
If x = tany, then y = arctanx, and so:

log(1+ix)-log(1-ix) = log[(1+ix)/(1-ix)]

= log[(cosy + isiny)/(cosy - isiny)]

= log[e^{2iy}]

= 2iy

:smile: = 2i.arctan(x). :smile:

(You see how, to prove anything with arctan(x), you always convert to x = tany?)
 
  • #11
You basically want to show that

[tex]\arctan (1/v) + \arctan (v) = \frac{\pi}{2}[/tex].

Draw a right angled triangle, with the smaller sides length 1 and v. What does [itex]\arctan v[/itex] represent here?
 

1. What makes a proof interesting?

An interesting proof is one that is unexpected or counterintuitive, yet logically sound. It may also involve creative and unique approaches to solving a problem.

2. How can a proof be considered challenging?

A challenging proof is one that requires a high level of mathematical knowledge and problem-solving skills. It may involve complex concepts or require multiple steps to reach a conclusion.

3. What are some common challenges in creating an interesting proof?

Some common challenges in creating an interesting proof include identifying and understanding the key concepts and assumptions, finding a logical and elegant argument, and ensuring that the proof is rigorous and complete.

4. How can I improve my ability to create interesting yet challenging proofs?

To improve your ability to create interesting yet challenging proofs, it is important to continually practice and develop your mathematical reasoning skills. This can involve studying different proof techniques, analyzing and understanding the logic behind existing proofs, and constantly challenging yourself to solve more difficult problems.

5. Why are interesting yet challenging proofs important in science?

Interesting yet challenging proofs are important in science because they help to advance our understanding of the natural world and solve complex problems. They also demonstrate the power and beauty of mathematics as a tool for discovery and innovation.

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