MATHEMATICAL FORMULA HANDBOOK

• Hyperbolic Function

1. cosh x = ½ (e^x + e^-x) = 1+ x² ⁄ 2! + x⁴ ⁄ 4! + .....
2. sinh x = ½ (e^x - e^-x) = x+x³ ⁄ 3! + x⁵ ⁄ 5! + .......
3. cosh ix = cos x
4. sinh ix = i sin x
5. tanh x = sinh x ⁄ cosh x
6. coth x = cosh x ⁄ sinh x
7. cosh² x = sinh² x = 1

Relations of the functions

1. sinh x = -sinh (−x)
2. cosh x = cosh (−x)
3. sinh x = 2 tanh (x⁄2) ⁄ (1 − tanh² (x⁄2)) = tanh x &frasl (√ 1 − tanh² x)
4. tanh x = √ (1 − sech² x)
5. coth x = √ (cosech² x + 1)
6. sinh(x ⁄ 2) = √[(cosh x -1) ⁄ 2]
7. tanh (x ⁄ 2) = (cosh x − 1) ⁄ sinh x = sinh x ⁄ (cosh x + 1)
8. sinh(2x) = 2sinh x cosh x
9. cosh(2x) = cosh² x + sinh² x − 1 = 1 + 2 sinh² x
10. sinh(3x) = 3sinh x + 4 sinh³ x
11. cosh(3x) = 4 cosh³ x − 3cosh x
12. tanh(3x) = (3tanh x +tanh³ x) ⁄ (1 + 3 tanh² x)
13. sinh(x ± y) = sinh x cosh y ± cosh x sinh y
14. cosh(x ± y) = cosh x cosh y ± sinh x sinh y
15. tanh(x ± y) = (tanh x ± tanh y) ⁄ (1 ± tanh x tanh y)
16. sinh x + sinh y = 2 sinh ½ (x+y) cosh ½ (x-y)
17. sinh x − sinh y = 2 cosh ½ (x+y) sinh ½ (x-y)
18. cosh x + cosh y = 2 cosh ½ (x+y) cosh ½ (x-y)
19. cosh x − cosh y = 2 sinh ½ (x+y) sinh ½ (x-y)
20. sinh x ± cosh x = [(1 ± tanh (x ⁄ 2)) ⁄ (1 ± tanh (x ⁄ 2))] = e ^(±x)
21. tanh x ± tanh y = [(sinh (x ± y)) ⁄ (cosh x cosh y)]
22. coth x ± coth y = ± [(sinh (x ± y)) ⁄ (sinh x sinh y)]

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