Interesting yet challenging proof (1 Viewer)

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

how can I show that ... arctan(1/v) = (π/2) - arctan(v) ???
 
well,try giving and playing with the variables untill you get a result.

More of a trial and error method.

Einstienear.
 
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?
 
1,341
3
Do v and n represent anything in particular?

Because your equation only works when you have certain values for v and n.
 
i knew this would confuse someone... that "n" you see is π, or pi... sorry. Does that help?
 
192
1
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.
 

tiny-tim

Science Advisor
Homework Helper
25,799
242

dynamicsolo

Homework Helper
1,649
4
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...)
 
Last edited:
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 :)
 

tiny-tim

Science Advisor
Homework Helper
25,799
242
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?)
 

Gib Z

Homework Helper
3,345
2
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?
 

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top