Queetion 912 The goal of these questions; is to introduce you to the hyperbolic trigonometric functions, and their properties. Analogiee with the trigonometric functions are made, and more will be given in M135. We dene the functions: a: _ :r a = a _ _= einh(:|:] = % \"1511(5) = i L Question 9 (18 Marks) a) State the domain and range of each of these functions. b) Sketch each of these functions. c) Show that cosh2 (x) ! sinh2 (x) = 1: The hyperbolic trigonometric functions are associated with hyperbolae (p2 ! q 2 = 1) in a similar manner that trigonometric functions are associated with circles (p2 + q 2 = 1): d) Show that cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y): Note the similarity with the analogous trigonometric identity. Question 10. (18 Marks) e) Evaluate the derivatives of each of these new functions, expressions your answer in terms of these three function only. f) Show that both cosh(x) and sinh(x) solve the second order dierential equation: & ' d dy = y: dx dx (Just verify that each function satises this equation). Note that cos(x) and sin(x) solve the second order dierential equation: & ' d dy = !y: dx dx g) Suppose that 0 ; by: tanh > = U: If I observe an object moving with speed U; (and rapidity >U relative to me) and this object observes a particle with speed V (and rapidity >V relative to the object) then the rapidity of the particle relative to me is given by: > = >U + >V ; that is, the relative rapidities are additive. Make use of the previous part of the question to determine the relative velocity, W; of the particle to me, in terms of U and V only. This is known as the relativistic velocity addition law. m) If U = 23 and V = 23 then what is W ? Note that your solution will reveal the surprising fact that: if you have a velocity of 2c 3 relative to me, and your friend has a velocity of 2c 3 relative to you, then your friend does not have a velocity of 4c 3 relative to me. 3