Hyperbolic Functions – Formulas and Problems

Hyperbolic Functions – Formulas and Problems

Formulas:

• $${{\cosh }^{2}}x-{{\sinh }^{2}}x=1$$,
• $$1-{{\tanh }^{2}}x={{\operatorname{sech}}^{2}}x$$,
• $${{\cosh }^{2}}x-1=\cos ec{{h}^{2}}x$$,
• $${{\sinh }^{-1}}x={{\log }_{e}}\left( x+\sqrt{{{x}^{2}}+1} \right)$$,
• $${{\cosh }^{-1}}x={{\log }_{e}}\left( x+\sqrt{{{x}^{2}}-1} \right)\,\,\,for\,\,\,x\ge 1$$,
• $${{\tanh }^{-1}}x=\frac{1}{2}\log \left( \frac{1+x}{1-x} \right)\,\,\,\,for\,\,\,\,x\in \left( -1,1 \right)$$,
• $${{\coth }^{-1}}x=\frac{1}{2}\log \left( \frac{x+1}{x-1} \right)\,\,\,\,\,for\,\,\,\left| x \right|>1$$,
• $$\sec {{h}^{-1}}x={{\log }_{e}}\left( \frac{1+\sqrt{1-{{x}^{2}}}}{x} \right)\,\,\,\,for\,\,\,x\in \left( 0,1 \right]$$,
• $$\cos ec{{h}^{-1}}x={{\log }_{e}}\left( \frac{1+\sqrt{1+{{x}^{2}}}}{x} \right)\,\,\,\,for\,\,\,x>0$$,
• $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\log }_{e}}\left[ \frac{1-\sqrt{1+{{x}^{2}}}}{x} \right]\,\,\,\,for\,\,\,x<0$$,
• $${{\sinh }^{-1}}x={{\cosh }^{-1}}\left( \sqrt{{{x}^{2}}+1} \right)=\cos ec{{h}^{-1}}\left( \frac{1}{x} \right)={{\tanh }^{-1}}\left( \frac{x}{\sqrt{1+{{x}^{2}}}} \right)$$,
• $${{\cosh }^{-1}}x={{\sinh }^{-1}}\left( \sqrt{{{x}^{2}}-1} \right)=\sec {{h}^{-1}}\left( \frac{1}{x} \right)={{\tanh }^{-1}}\left( \frac{\sqrt{{{x}^{2}}-1}}{x} \right)$$.

Examples 1: If sinhx = ¾ find (i) cosh2x and (ii) sinh2x.

Solution: Given that

sinhx = ¾

(∵cosh²x – sinh²x = 1)

cosh²x = 1 + sinh²x

$${{\cosh }^{2}}x=1+\left( \frac{9}{16} \right)$$,

$${{\cosh }^{2}}x=\frac{25}{16}$$,

Coshx = 5/4.

(i) Sinh2x = 2 sinhx coshx

$$2\times \frac{3}{4}\times \frac{5}{4}=\frac{15}{8}$$,

(ii) cosh 2x = cosh² x + sinh² x.

$$\frac{25}{16}+\frac{9}{16}=\frac{34}{16}=\frac{17}{8}$$.

Example 2: If sin hx = 3, x = log (3 – √10).

Solution: Given that sin hx = 3

x = log (3 – √10)

sin hx = 3

x = sinh⁻¹ (3)

sinh⁻¹ y = cos [y + √(y² + 1)]

sinh⁻¹ 3 = cos [3 + √(3² + 1)]

sinh⁻¹ 3 = cos [3 + √(9 + 1)]

sinh⁻¹ 3 = cos (3 + √10).