MOOC MAT438/ UiTM
Definition and Identities of Hyperbolic Functions
Proving the Hyperbolic Function using Identities and Definition
Solving the Hyperbolic Equations using Identities and Definition
Derivatives of Hyperbolic Functions
Integration of Hyperbolic Functions
MOOC MAT438/ UiTM
Example 1
Show that ℎ2 + 1 = ℎ2 .
Solution :
ℎ2 + 1 = 1 ℎ2 ℎ2 − ℎ2 = 1
ℎ2 + ℎ2 ∴ ℎ2 = 1 + ℎ2
1 + ℎ2
= ℎ2
ℎ2
= ℎ2
= ℎ2 # Shown !
MOOC MAT438/ UiTM
Example 2
ℎ
Verify the hyperbolic identity : ℎ − ℎ = ℎ + 1
Solution : ℎ ℎ − 1
= ℎ ÷ ℎ
ℎ ℎ ℎ − 1
ℎ ℎ
= ℎ ∙ ℎ − 1
ℎ − ℎ = ℎ − 1
ℎ ℎ
ℎ2
ℎ = ℎ − 1
= ℎ − 1
ℎ
ℎ2 ℎ2 − ℎ2 = 1
= ℎ − 1 ∴ ℎ2 − 1 = ℎ2
ℎ2 − 1 = ℎ + 1 # Verified ! − = + −
= ℎ − 1 ℎ2 − 1 = ℎ 2 − 1 2
ℎ + 1 ℎ − 1 = ℎ + 1 ℎ − 1
= ℎ − 1
MOOC MAT438/ UiTM
MOOC MAT438/ UiTM
Definition and Identities of Hyperbolic Functions
Proving the Hyperbolic Function using Identities and Definition
Solving the Hyperbolic Equations using Identities and Definition
Derivatives of Hyperbolic Functions
Integration of Hyperbolic Functions
MOOC MAT438/ UiTM
Example 3
Use the definitions of hyperbolic function to prove that ℎ2 − ℎ2 = 1
Solution : + − 2 − − 2 + = + + − = − −
2 −2 = + + = − +
ℎ2 − ℎ2 =
+ − 2 = 2 +2 − + − 2 − − 2 = 2 −2 − + − 2
+ − 2 − − 2 = 2 + 2 + −2 = 2 − 2 + −2
=4−4
− = −
= 0
+ − 2 − − − 2 =1
=4
2 + 2 + −2 − 2 − 2 + −2
=4
2 + 2 + −2 − 2 + 2 − −2 4 =1 # Proven !
= =4
MOOC MAT438/ UiTM 4
Example 4
Use the definitions of hyperbolic function to show that 1 − ℎ2 = ℎ2
Solution : + = + + − = − −
= + + = − +
1 − ℎ2 = + − 2 − − 2
+ − − + − + − 2 = 2 +2 − + − 2 − − 2 = 2 −2 − + − 2
= 2 + 2 + −2 = 2 − 2 + −2
+ − 2 − − 2
= + − 2 − + − 2
+ − 2 − − − 2
= + − 2
2 + 2 + −2 − 2 − 2 + −2
= + − 2
2
2 + 2 + −2 − 2 + 2 − −2 4 22 = 2 # Shown !
= = + − 2 = + − 2 + − = ℎ2
MOOC MAT438/ UiTM + − 2
MOOC MAT438/ UiTM