FAC1004 Tutorial 8 — Hyperbolic Functions
Centre for Foundation Studies in Science
University of Malaya
FAC1004 Advanced Mathematics 2
Topic: Hyperbolic Functions
Question 1
Given that $\sinh x = \frac{5}{12}$, find the values of:
(a) $\cosh x$
(b) $\text{csch} x$
(c) $\tanh x$
(d) $\sinh 2x$
(e) $\cosh 5x$
Question 2
Show that $x = \frac{1}{2}\ln 3$ if $\tanh x = \frac{1}{2}$.
Question 3
Find all possible values of $\sinh x$ for which $12\cosh^2 x + 7\sinh x = 24$.
Question 4
Find $\frac{dy}{dx}$ for the following functions.
(a) $y = \sinh(4x - 8)$
(b) $y = \cosh(x^4)$
(c) $y = \coth(\ln x)$
(d) $y = \ln(\tanh 2x)$
(e) $y = \sinh^3(2x)$
Question 5
Differentiate the following with respect to $x$.
(a) $y = x^4 \cosh(3x^2 - 8x)$
(b) $y = \cosh(\sin^{-1} x^4)$
(c) $y = \coth(\ln x) \cos^{-1}(2x^3)$
(d) $y = \frac{\ln(\sin 2x)}{\coth(x^2)}$
(e) $y = \frac{\sinh^3(2x)}{\cosh^2(5x^{-3} + x)}$
Question 6
Evaluate the following integrals.
(a) $\int \sinh x \cosh(x^6) dx$
(b) $\int \cosh(2x - 3) dx$
Question 7
Evaluate the exact value for each of the following.
(a) $\tanh^{-1} 0$
(b) $\cosh^{-1} \pi$
(c) $\sinh^{-1}\left(\frac{\pi}{4}\right)$
Question 8
Prove that $\sinh^{-1} x = \ln\left[x + \sqrt{1 + x^2}\right]$.
Question 9
Find the value of $x$ for:
(a) $\sinh^{-1}\frac{3}{4} + \sinh^{-1} x = \sinh^{-1}\frac{4}{3}$
(b) $\cosh^{-1} x + \cosh^{-1}\frac{3}{4} = -\cosh^{-1}\frac{4}{3}$
(c) $\tanh^{-1}\frac{2}{5} - \tanh^{-1} x = \sinh^{-1}\frac{3}{5}$
Key Concepts Covered
- Hyperbolic Functions — Definitions and basic properties
- Hyperbolic Identities — Fundamental relationships
- Osborn's Rule — Converting trig to hyperbolic identities
- Derivatives of Hyperbolic Functions — d/dx(sinh x) = cosh x, etc.
- Integrals of Hyperbolic Functions — Basic antiderivatives
- Inverse Hyperbolic Functions — Definitions and logarithmic forms