I this difficult geometry question I've been stuck on for a while, I can do parts of it but think I've gone wrong at some point and don't know how to continue, wondering if I could get any help.
The hyperbolic lines l, m and n defined by:
l: = | z - 1 | = 2, m: = Re(z) = 2 , n = | z - 3 | = 2root3
form a hyperbolic triangle with vertices: A := m \ n, B := l\ n and C := l \ m
(i)Find the complex numbers which represent A, B and C
I found A, B and C.
A: x=2, y= root11
B: x = 3 y = root3
C: x=2, y= root3
But I'm not 100% sure I've done this correct.
(ii) Draw a diagram to scale of the hyperbolic lines l, m and n. Label their points
of intersection and their intersections with the real axis with the appropriate
letters and complex numbers
This part is simply plotting the points found in (i) which I've done.
(iii) Two of the interior angles are denoted: x at B and y at C, give exact expressions for cos (y), sin(y), cos(x). Calculate the angles x and y to 3 places of decimals.
This is the question I'm now stuck on as the angle at C should be 90 degrees if my points are correct? Not sure what I've done wrong I'm sure I've made a mistake.
These next two questions I have no idea about yet, and would appreciate a steer but I havn't attempted them too much yet.
(iv) Here length refers to hyperbolic length (i.e distance). Let a denote the length of the h-line segment BC and b the length of the h-line segment AC. Calculate a and b exactly in terms of natural logarithms and also approximately, to 4 decimal places.
(v) Using the exact answers in (iii) and (iv) verify exactly, for the given hyperbolic
triangle, the following identity (partial marks may be obtained for an
approximate verication): cos(y)cosh(a) = sinh(a)coth(b) - sin(y)cot(x)
any help at all would be greatly appreciated.