Functions involving certain combinations of e* and ex occur so frequently in applied mathematics that they warrant special study. These functions, called the hyperbolic functions, have properties similar to those of trigonometric functions. Because of this, they are called the hyperbolic sine (sinh), the hyperbolic cosine (cosh), the hyperbolic tangent(tanh) functions. The hyperbolic functions are defined for all real x as: sinh x = ex - e-x 2 exe cosh x = tanh x = 2 sinh x cosh x ex - e-x = ex + e-x Hyperbolic functions are related to a hyperbola in much the same way as the trigonometric functions (sometimes called circular functions) are related to the circle. Just as any point P on the unit circle has coordinates (cost, sin t), a point on the hyperbola x y = 1 has the coordinates (cosht, sinh t). (cost, sin f) t -0.5 (cosht, sinh t) a) Establish the following identities. (Use LHS/RHS approach.) i. cosh2 x sinh2 x 1 (2 pts) ii. sinh(x) = = sinhx (1 pt) iii. sinh(A + B) = sinh A cosh B + cosh A sinh B (2 pts) b) i. Establish the following differentiation formulae -sinh x = coshx (1 pt) c) dx ii. cosh x=sinh x (1 pt) dx A cable is suspended between two poles of the same height that are 20m apart. If the poles are placed at (-10, 0) and (10, 0), the equation that models the height of the cable is x y = 10 cosh+15 10 (Interestingly this minimises the gravitational potential energy of the cable as a whole.) Find the acute angle at which the cable meets the pole. Answer in degrees correct to two decimal places. A diagram will be useful here. (4 pts)