**Problem 1C-10*:** Hyperbolic functions and hyperbolic orbit

Hyperbolic functions sinh(z) and cosh(z) are defined as

(a) Prove sinh(-z)=-sinh(z) and cosh(-z)=cosh(z).

(b) Prove cosh^{2}(z)-sinh^{2}(z) = 1.

(c) Prove sinh(A+B) = sinh(A)·cosh(B)+cosh(A)·sinh(B).

(d) Prove cosh(A+B) = cosh(A)·cosh(B)+sinh(A)·sinh(B).

(e) If y(t) = A·cosh(ωt+α) and x(t) = B·sinh(ωt+β), what is the trajectory of the motion?

(f) Calculate the velocity and the acceleration vectors for this motion.

**Solution:**

(a)

(b)

(c)

(d)

(e)

so the trajectory is a hyperbola.

(f) Let **r**(t) = (x(t), y(t)) = (B·sinh(ωt+α), A·cosh(ωt+α)).
Then

This means that the particle moves faster and faster as t increases and the particle moves away from the origin. For the acceleration vector we have

so |

so

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