Inverse Hyperbolic Functions Practice Problems

In mathematics, the inverse functions of hyperbolic functions are referred to as inverse hyperbolic functions or area hyperbolic functions. In this article, we will learn what inverse hyperbolic functions are and practice many questions.

What are Inverse Hyperbolic Functions?

In mathematics, the inverse functions of hyperbolic functions are referred to as Inverse Hyperbolic Functions or area hyperbolic functions. There are six inverse hyperbolic functions, namely, inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent functions. These functions are depicted as sinh^-1 x, cosh^-1 x, tanh^-1 x, csch^-1 x, sech^-1 x, and coth^-1 x. With the help of an inverse hyperbolic function, we can find the hyperbolic angle of the corresponding hyperbolic function.

Important Formulas

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)𝑔(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))