1—Basic Stuff 24Problems1.1 What is the tangent of an angle in terms of its sine? Draw a triangle and do this in one line.
1.2 Derive the identities forcosh^2 θ−sinh^2 θand for 1 −tanh^2 θ, Equation ( 3 ).
1.3 Derive the expressions forcosh−^1 y,tanh−^1 y, andcoth−^1 y. Pay particular attention to the domains and
explain why these are valid for the set ofythat you claim. What issinh−^1 (y) + sinh−^1 (−y)?
1.4 The inverse function has a graph that is the mirror image of the original function in the 45 ◦liney=x.
Draw the graphs of all six of the hyperbolic functions and all six of the inverse hyperbolic functions, comparing
the graphs you should get to the functions derived in the preceding problem.
1.5 Evaluate the derivatives ofcoshx,tanhx, andcothx.
1.6 What are the derivatives,dsinh−^1 y
/
dyanddcosh−^1 y/
dy?1.7 Find formulas forcosh 2yandsinh 2yin terms of hyperbolic functions ofy. The second one of these should
take at most two lines. Maybe the first one too.
1.8 Do a substitution to evaluate the integral (a) simply. Now do the same for (b)
(a)∫
dt
√
a^2 −t^2(b)∫
dt
√
a^2 +t^21.9 Sketch the two integrands in the preceding problem. For the second integral, if the limits are 0 andxwith
x a, then before having done the integral, estimateapproximately what the value of this integral should be.
(Sayx= 10^6 aorx= 10^60 a.) Compare your estimate to the exact answer that you just found to see if they
match in any way.
1.10 Fill in the steps in the derivation of the Gaussian integrals, Eqs. ( 7 ), ( 8 ), and ( 10 ). In particular, draw
graphs of the integrands to show why Eq. ( 7 ) is so.