Calculus- Early Transcendentals, 2021a

68 Functions
(a) sin −1 ( √ 3/2) (b) cos −1 (− √ 2/2)
Exercise 2.6.2 Compute the following:
(a) sin −1 (sin(π/4))
(b) sin −1 (sin(17π/3))
(c) cos ( cos −1 (1/3) )
(d) tan ( cos −1 (−4/5) )
Exercise 2.6.3 Rewrite the expression tan ( cos −1 x ) without trigonometric functions. What is the domain
of this function?
2.7 Hyperbolic Functions
The hyperbolic functions appear with some frequency in applications, and are quite similar in many respects
to the trigonometric functions. This is a bit surprising given our initial definitions.
Definition 2.29: Hyperbolic Sine and Cosine
The hyperbolic cosine is the function
and the hyperbolic sine is the function
coshx = ex + e −x
,
2
sinhx = ex − e −x
.
2
Notice that cosh is even (that is, cosh(−x) =cosh(x)) while sinh is odd (sinh(−x) =−sinh(x)), and
coshx + sinhx = e x .Also,forallx, coshx > 0, while sinhx = 0 if and only if e x − e −x = 0, which is true
precisely when x = 0.
Theorem 2.30: Range of Hyperbolic Cosine
The range of coshx is [1,∞).
Proof. Let y = coshx. Wesolveforx:
y = ex + e −x
2
2y = e x + e −x
2ye x = e 2x + 1
0 = e 2x − 2ye x + 1