Definitions of hyperbolic sine and cosine Complete the

Chapter 6, Problem 112

(choose chapter or problem)

Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x2 - y2 = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as a. Explain why twice the area of the shaded region is given by t = 2 # a 1 2 xy - L x 1 2z2 - 1 dzb = x2x2 - 1 - 2 L x 1 2z2 - 1 dz. b. In Chapter 7, the formula for the integral in part (a) is derived: L 2z2 - 1 dz = z 2 2z2 - 1 - 1 2 ln _ z + 2z2 - 1 _ + C. Evaluate this integral on the interval 31, x4, explain why the absolute value can be dropped, and combine the result with part (a) to show that t = ln 1x + 2x2 - 12. c. Solve the final result from part (b) for x to show that x = et + e-t 2 . d. Use the fact that y = 2x2 - 1 in combination with part (c) to show that y = et - e-t 2 .