Question

1. Starting with the equation a tanh y show that, for real r, y and |rl < 1, y-tanh-1 2. Compute the following derivative dtanh (sin() 3. Assume θ is a real number. Then use Euler's formula eie-cos θ + isin θ to show that coth(i0)-icot(e) 4. Use the definitions to obtain an equation for cosh(3x) in terms of cosh(x) and sinh(x) and their various products (e.g., cosh*(z), cosh(x) sinh3(x) etc.). Do not use the double-angle formula such as cosh(u+ v)-cosh(u) cosh(v) + sinh() sinh

Answer 1

x = tan hy x = tan hy = sin hy/cos hy = e^y - e^-y/2/e^y + e^-y/2 = e^y - e^-y/e^y + e^-y Implies x = (e^y - e^-y/e^y + e^-y) e^y/e^y (e^2y + 1) x = e^2y - 1/e^2y + 1 (e^2y/+ 1) xe^2y + x = e^2y - 1 Implies xe^2y - e^2y = -1 - x = -(1 + x) Implies e^2y - xe^2y = (1 + x) Implies e^2y (1 - x) = 1 + x Implies e^2y = 1 + x/1 - x log_e e^2y = log (1 + x/1 - x) Implies 2y = log_e (1 + x/1 - x) therefore y = tan h^-1 (x) = 1/2 log_e (1 + x/1 - x) \r\n (2) d/dx [tan h^-1 (sin x)] = 1/1 - (sin x)^2 cos x = cos x/1 - sin^2 x = cos x/cos^2 x = 1/cos x = sec x \r\n (4) cos hx = e^x + e^-x/2: sin hx = e^x - e^-x/2 cos h (3x) = 4 cos h^3 x - 3 cos hx = 4 (e^x + e^-x/2)^3 - 3 (e^x + e^-x/2)