Hello everybody and thanks for reading.
I have been trying to solve the following problem without success, so I am looking for some advices..
I have the following equation
arccos(x) = arctan( root(1 - x^2) / x)
So I have to prove if that 'identity' is valid and say for what values of x is valid.
Even when I am able to construct the triangle from the data, I don't know how to find the valid values for x.
So far I have constructed a triangle with opposite root(1-x^2), adjacent x and hypotenuse 1.
I also restricted the domain so 1-x^2 >= 0, x != 0 and -1 <= x <= 1 (for the root, denominator, and arccos respectively).
I also tried to prove the identity using the Pythagorean theorem from the data of the arctan ( opposite and adjacent), and I get that the hypotenuse is 1, which is the same the arccos says.
But I have seen the plot of both functions and I realized they are only equal for 0 < x <= 1 and I don't know how I could get there. Intersecting the restrictions I said I only get [-1,1] - {0}.
I have even thought I should intersect the ranges of those functions, but I don't think that is valid.
Thanks in advance and sorry for the long post
Yes, that is what I did.. But if you try the identity for, say, -1, it is not true... So what I need to do is to find (hopefully algebraically) the values that make that identity true.
I constructed the triangle doing the same as you did, but I don't know how should i proceed.
Thank you very much
Let denote . We have the following facts.
If , then (1)
If , then (2)
From (2), if , then (3)
The function arctan is the inverse of tan on (4)
(5)
From (3) and (4), if , then , so (6)
So, if , then
by (1)
by (3).
Therefore,
by (6). For we simialrly have and , so .
Thank you very much for your answers.
I wanted something like what emakarov posted, so it helped a lot. I tried to put it in simple terms so concluded:
- tan (theta) = opposite / adjacent, but as we are talking about lengths, negative values for any of both have no sense. In this case, x was in the denominator, si if it is negative it has no sense.
- So tan(theta) = sin(theta) / cos(theta) . As the numerator is always non-negative, theta must be an angle between [0, PI] so its sin is positive. Then, for theta to be positive, x >= 0, but as it can't be 0, it must be between (0, 1], considering the restrictions on the domain.
Thank you very much