2.1 Proving Hyperbolic Functions (using Identities & Definition)

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