1. Hyperbolic functions

2. FM Hyperbolic functions: Identities and Equations
KUS objectives
BAT use identities for the hyperbolic functions; solve equations using the definitions and identities for hyperbolic functions
Starter: Use the addition formula to show that
sin 2𝑥 =2 sin 𝑥 cos 𝑥
cos 2𝑥 =1−2 𝑠𝑖𝑛 2 𝑥
cos 4𝑥 = 𝑐𝑜𝑠 2 2𝑥− 𝑠𝑖𝑛 2 2𝑥

3. tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
Notes (reminder) Hyperbolic functions have several properties in common with trigonometric functions, but they are defined in terms of exponential functions
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
‘shine x’
‘coshine x’
There are corresponding reciprocal functions
cosech 𝑥 ≡ 2 𝑒 𝑥 − 𝑒 −𝑥
sech 𝑥 ≡ 2 𝑒 𝑥 + 𝑒 −𝑥
coth 𝑥 ≡ 1 tanh 𝑥 ≡ 𝑒 2𝑥 +1 𝑒 2𝑥 −1
‘cosheck x’
‘sheck x’
These definitions can simply be stated but need to be memorised

4. WB C1 Prove that 𝑎) 𝑐𝑜𝑠ℎ 2 𝑥− 𝑠𝑖𝑛ℎ 2 𝑥≡1
𝑏) sinh 𝐴+𝐵 = sinh 𝐴 cosh 𝐵 + cosh 𝐴 sinh 𝐵
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
𝑎) 𝐿𝐻𝑆= 𝑒 𝑥 + 𝑒 −𝑥 − 𝑒 𝑥 − 𝑒 −𝑥
= 𝑒 2𝑥 +2+ 𝑒 −2𝑥 4 − 𝑒 2𝑥 −2+ 𝑒 −2𝑥 4 = 4 4 = QED
𝑏) 𝑅𝐻𝑆= 𝑒 𝐴 − 𝑒 −𝐴 𝑒 𝐵 + 𝑒 −𝐵 𝑒 𝐴 + 𝑒 −𝐴 𝑒 𝐵 − 𝑒 −𝐵 2
= 𝑒 𝐴+𝐵 + 𝑒 𝐴−𝐵 − 𝑒 −𝐴+𝐵 − 𝑒 −𝐴−𝐵 𝑒 𝐴+𝐵 − 𝑒 𝐴−𝐵 + 𝑒 −𝐴+𝐵 − 𝑒 −𝐴−𝐵 4
= 2 𝑒 𝐴+𝐵 −2 𝑒 −𝐴−𝐵 4
= 𝑒 𝐴+𝐵 − 𝑒 −𝐴−𝐵 2
= sinh (𝐴+𝐵) =𝐿𝐻𝑆 QED

5. tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 𝑐𝑜𝑠ℎ 2 𝑥− 𝑠𝑖𝑛ℎ 2 𝑥≡1
Notes the addition formulae for hyperbolic functions are
sinh 𝐴±𝐵 = sinh 𝐴 cosh 𝐵 ± cosh 𝐴 sinh 𝐵
cosh 𝐴±𝐵 = cosh 𝐴 cosh 𝐵 ± sinh 𝐴 sinh 𝐵
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥
𝑐𝑜𝑠ℎ 2 𝑥− 𝑠𝑖𝑛ℎ 2 𝑥≡1

6. WB C2 a) prove that cosh 2𝑥 =1+2 𝑠𝑖𝑛ℎ 2 𝑥
b) Write the hyperbolic identity corresponding to
cos 2𝑥 =2 𝑐𝑜𝑠 2 𝑥−1
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
𝑎) 𝑅𝐻𝑆= 𝑒 𝑥 − 𝑒 −𝑥
OSBORNS RULE
sin 𝑥 → sinh 𝑥
cos 𝑥 → cosh 𝑥
Replace any product of two sin terms by minus the product of sinh terms
e.g. sinAsinB → - sinh A sinhB
e.g. tanAtanB → - tanh A tanhB
=1+2 𝑒 2𝑥 −2+ 𝑒 −2𝑥 4
=1−1+ 𝑒 2𝑥 + 𝑒 −2𝑥 2
= cosh 2𝑥 =LHS QED
𝑏) cos 2𝑥 =2 𝑐𝑜𝑠ℎ 2 𝑥−1

7. WB C3 Given that sinh 𝑥 = 3 4 find the exact value of
𝑎) cosh 𝑥 𝑏) tanh 𝑥 𝑐) sinh 2𝑥 𝑑) cosℎ 2𝑥
𝑏) tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥
𝑎) 𝑐𝑜𝑠ℎ 2 𝑥− 𝑠𝑖𝑛ℎ 2 𝑥≡1
𝑐𝑜𝑠ℎ 2 𝑥=1+ 𝑠𝑖𝑛ℎ 2 𝑥= = 25 16
= 3/4 5/4 = 3 5
cosh 𝑥 = 5 4
𝑐) sinh 2𝑥 ≡2 sinh 𝑥 cosh 𝑥
𝑑) cosh 2𝑥 ≡1+2 𝑠𝑖𝑛ℎ 2 𝑥
= = 17 8
=2× 3 4 × 5 4 = 15 8

8. 3 𝑒 𝑥 −3 𝑒 −𝑥 − 𝑒 𝑥 − 𝑒 −𝑥 =7 collect and multiply through by 𝑒 𝑥
WB C4a Solving equations solve each equation for all real values of x, give answers as natural logarithms where appropriate
𝑎) 6 sinh 𝑥 −2 cosh 𝑥 =7
𝑏) 2 𝑐𝑜𝑠ℎ 2 𝑥−5 sinh 𝑥 =5
𝑐) cosh 2𝑥 −5 𝑐𝑜𝑠ℎ 𝑥 +4=0
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
𝑎) 6 𝑒 𝑥 − 𝑒 −𝑥 2 −2 𝑒 𝑥 + 𝑒 −𝑥 2 =7
3 𝑒 𝑥 −3 𝑒 −𝑥 − 𝑒 𝑥 − 𝑒 −𝑥 =7 collect and multiply through by 𝑒 𝑥
2 𝑒 2𝑥 −7 𝑒 𝑥 −4=0
𝑒 𝑥 =− 1 2 , but 𝑒 𝑥 >0
𝑒 𝑥 −4 𝑒 𝑥 +1 =0
𝑥= ln 4

9. 𝑥=𝑎𝑟𝑠𝑖𝑛ℎ − 1 2 = ln − 1 2 + − 1 2 2 +1 = ln − 1 2 + 5 2
WB C4 b Solving equations solve each equation for all real values of x, give answers as natural logarithms where appropriate
𝑏) 2 𝑐𝑜𝑠ℎ 2 𝑥−5 sinh 𝑥 =5
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
𝑏) use the identity 𝑐𝑜𝑠ℎ 2 𝑥=1+ 𝑠𝑖𝑛ℎ 2 𝑥
𝑠𝑖𝑛ℎ 2 𝑥 −5 sinh 𝑥 =5
2 𝑠𝑖𝑛ℎ 2 𝑥−5 sinh 𝑥 −3=0
arsinℎ 𝑥 = ln 𝑥+ 𝑥 𝑥∈𝑅
arcosh 𝑥 = ln 𝑥+ 𝑥 2 −1 , 𝑥≥1
2 sinh 𝑥 +1 sinh 𝑥 −3 =0
sinh 𝑥 =− 1 2 , 3
𝑥=𝑎𝑟𝑠𝑖𝑛ℎ − 1 2 = ln − − = ln − Or
𝑥=𝑎𝑟𝑠𝑖𝑛ℎ (3)= ln = ln

10. 2 𝑐𝑜𝑠ℎ 2 𝑥−1 −5 cosh 𝑥 +4=0 2 𝑐𝑜𝑠ℎ 2 𝑥−5 cosh 𝑥 +3=0
WB C4 c Solving equations solve each equation for all real values of x, give answers as natural logarithms where appropriate
𝑐) cosh 2𝑥 −5 𝑐𝑜𝑠ℎ 𝑥 +4=0
sinh 𝑥 ≡ 𝑒 𝑥 − 𝑒 −𝑥 2
cosh 𝑥 ≡ 𝑒 𝑥 + 𝑒 −𝑥 2
tanh 𝑥 ≡ sinh 𝑥 cosh 𝑥 ≡ 𝑒 2𝑥 −1 𝑒 2𝑥 +1
𝑐) use the identity cosh 2𝑥 =2 𝑐𝑜𝑠ℎ 2 𝑥−1
2 𝑐𝑜𝑠ℎ 2 𝑥−1 −5 cosh 𝑥 +4=0
2 𝑐𝑜𝑠ℎ 2 𝑥−5 cosh 𝑥 +3=0
arsinℎ 𝑥 = ln 𝑥+ 𝑥 𝑥∈𝑅
arcosh 𝑥 = ln 𝑥± 𝑥 2 −1 , 𝑥≥1
2 cosh 𝑥 −3 cosh 𝑥 −1 =0
cosh 𝑥 = 3 2 , 1
𝑥=𝑎𝑟𝑐𝑜𝑠ℎ = ln ± −1 = or
𝑡ℎ𝑖𝑛𝑘 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝑎𝑙 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 cosh 𝑥 Or
𝑥=𝑎𝑟𝑐𝑜𝑠ℎ (1)= ln −1 = ln 1 =0

11. One thing to improve is –
KUS objectives
BAT use identities for the hyperbolic functions; solve equations using the definitions and identities for hyperbolic functions
self-assess
One thing learned is –
One thing to improve is –

12. END