Function	Formula
sinh-1 x	ln[x + √(x2 + 1)]
cosh-1x	ln[x + √(x2 – 1)
tanh-1 x	½ ln[(1 + x)/(1 – x)]
csch-1 x	ln[(1 + √(x2 + 1)/x]
sech-1 x	ln[(1 + √(1 – x2)/x]
coth-1 x	½ ln[(x + 1)/(x – 1)]

Inverse Hyperbolic Functions Practice Questions with Solution

Problem 1: Find the derivative of y=cosh^-1(3x).

Solution:

Given y = cosh^-1(3x)

To find the derivative, we use the chain rule and the derivative of cosh^-1(3x):

dy/dx​ = d/dx​cosh^-1(3x)

dy/dx ​= 3/√{(3x)² – 1​}

dy/dx ​= 3/√(9x² – 1​)

Problem 2: Simplify tanh⁡^-1(x) + tanh⁡^-1(y)

Solution:

Use addition formula for inverse hyperbolic tangent:

tanh^-1(x) + tanh^-1(y) = tanh^-1 (x+y)/(1+xy​)

Thus,

= tanh^-1(x) + tanh^-1(y)

= tanh^-1 (x+y)/(1+xy​)

Problem 3: Find the derivative. y = -8coth^-1(21x³)

Solution:

y′ = -8[​1/1{-(21x​^3​​)​²​​​​​​](63x^​2​​)}]

y′ = -504x²/1 – 441x⁶

y′ = -504x² – 441x⁶

Problem 4: Differentiate R(t) = tan(t) + t²csch(t).

Solution:

R(t) = tan(t) + t²csch(t)

Differentiating

R′(t) = sec²(t) + 2tcsch(t) – t²csch(t)coth(t)

Problem 5: Differentiate f(x) = sinh(x) + 2cosh(x) – sech(x).

Solution:

f(x) = sinh(x) + 2cosh(x) – sech(x)

Differentiating

f′(x) = cosh(x) + 2sinh(x) + sech(x)tanh(x)

Problem 6: Solve for x: cosh^⁡-1(x) = 2

Solution:

Given cosh⁡^-1(x) = 2

x = cosh(2)

We know that,

cosh(y) = (e^y + e^-y)/2

Putting y = 2

cosh(2) = (e² + e^-2)/2

Inverse Hyperbolic Functions: Worksheet

Q1. Differentiate the given function : g(z)=z+1/tanh(z)?(?).

Q2. Solve sin⁡^-1(x) = π/4

Q3. Solve sin⁡(arcsin⁡(x)) = 0.5

Q4. Find arcsin⁡(sin⁡(θ)) for θ = 3π/4

Q5. Solve arcsin⁡(sin⁡(5π/6))

Q6. Simplify cosh⁡(arsinh(x))

Frequently Asked Questions

What is Inverse Hyperbolic Function?

The inverse form of a hyperbolic function is called the inverse hyperbolic function.

What are the Domains and Ranges of the Sin Inverse Hyperbolic Functions?

Domains and ranges of the sin inverse hyperbolic functions are:

• Domain: (-∞, ∞)
• Range: (-∞, ∞)

What are the Domains and Ranges of the Cos Inverse Hyperbolic Functions?

Domains and ranges of the cos inverse hyperbolic functions

• Domain: [1, ∞)
• Range: [0, ∞)

What are Applications of Inverse Hyperbolic Functions?

Inverse hyperbolic functions are used in various fields such as:

• Mathematics: Solving integrals, differential equations, and in the study of complex numbers.
• Physics: Describing the shape of catenary curves, and in relativistic velocity addition.
• Engineering: Signal processing and control theory.