Hyperbolic Functions Practice Problems is curated to help students understand and master the concepts of hyperbolic functions. These problems cover basic operations and applications, providing a comprehensive approach to learning. By practicing these problems, students can strengthen their understanding of sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), and other hyperbolic functions, which are essential in fields like engineering, physics, and hyperbolic geometry

Hyperbolic functions are mathematical functions that are similar to trigonometric functions (like sine and cosine), but they’re based on hyperbolas instead of circles. They’re named sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), and so on.

In this article we have covered various Hyperbolic Functions Formulas and Practice Questions on Hyperbolic Functions.

Hyperbolic Functions Formulas

Various Hyperbolic Functions Formulas are:

Formula: sinh(x) = (e^x – e^(-x))/2

Formula: cosh(x) = (e^x + e^(-x))/2

Formula: tanh(x) = sinh(x) / cosh(x) = (e^x – e^(-x)) / (e^x + e^(-x))

Integrals Formulas:

• ∫ sinh(x) dx = cosh(x) + C
• ∫ cosh(x) dx = sinh(x) + C
• ∫ tanh(x) dx = ln(cosh(x)) + C

Important Identities:

• cosh²(x) – sinh²(x) = 1
• 1 – tanh²(x) = sech²(x)
• sinh(2x) = 2sinh(x)cosh(x)
• cosh(2x) = cosh²(x) + sinh²(x)

Reciprocal Identities:

• coth(x) = 1 / tanh(x)
• sech(x) = 1 / cosh(x)
• csch(x) = 1 / sinh(x)

Quotient Identity:

• tanh(x) = sinh(x) / cosh(x)

Double Angle Formulas:

• sinh(2x) = 2sinh(x)cosh(x)
• cosh(2x) = cosh²(x) + sinh²(x) = 2cosh²(x) – 1 = 1 + 2sinh²(x)
• tanh(2x) = (2tanh(x)) / (1 + tanh²(x))

Addition Formulas:

• sinh(x + y) = sinh(x)cosh(y) + cosh(x)sinh(y)
• cosh(x + y) = cosh(x)cosh(y) + sinh(x)sinh(y)
• tanh(x + y) = (tanh(x) + tanh(y)) / (1 + tanh(x)tanh(y))

Subtraction Formulas:

• sinh(x – y) = sinh(x)cosh(y) – cosh(x)sinh(y)
• cosh(x – y) = cosh(x)cosh(y) – sinh(x)sinh(y)
• tanh(x – y) = (tanh(x) – tanh(y)) / (1 – tanh(x)tanh(y))

Half Angle Formulas:

• sinh(x/2) = ±√((cosh(x) – 1) / 2)
• cosh(x/2) = √((cosh(x) + 1) / 2)
• tanh(x/2) = (cosh(x) – 1) / sinh(x) = sinh(x) / (cosh(x) + 1)

Power Reduction Formulas:

• sinh²(x) = (cosh(2x) – 1) / 2
• cosh²(x) = (cosh(2x) + 1) / 2

Sum and Difference of Hyperbolic Functions:

• sinh(x) + sinh(y) = 2sinh((x+y)/2)cosh((x-y)/2)
• sinh(x) – sinh(y) = 2cosh((x+y)/2)sinh((x-y)/2)
• cosh(x) + cosh(y) = 2cosh((x+y)/2)cosh((x-y)/2)
• cosh(x) – cosh(y) = 2sinh((x+y)/2)sinh((x-y)/2)

Hyperbolic Functions Practice Problems : Solved

1. Simplify: sinh²(x) – cosh²(x)

We know that: cosh²(x) – sinh²(x) = 1

We can rearrange this equation by subtracting cosh²(x) from both sides

-sinh²(x) = 1 – cosh²(x)

Now, multiply both sides by -1

sinh²(x) = -1 + cosh²(x)

Finally, subtract cosh²(x) from both sides

sinh²(x) – cosh²(x) = -1

Therefore, the simplified expression is -1

2. Find the derivative of tanh(x).

tanh(x) = sinh(x) / cosh(x)

To find the derivative, we’ll use the quotient rule:

d/dx[tanh(x)] = (cosh(x) × d/dx[sinh(x)] – sinh(x) × d/dx[cosh(x)]) / cosh²(x)

We know that the derivative of sinh(x) is cosh(x), and the derivative of cosh(x) is sinh(x)

Let’s substitute these:

d/dx[tanh(x)] = {cosh(x) × cosh(x) – sinh(x) × sinh(x)}/cosh²(x)

Now we can factor out cosh²(x) from the numerator:

d/dx[tanh(x)] = (cosh²(x) – sinh²(x)) / cosh²(x)

Recall the fundamental hyperbolic identity: cosh²(x) – sinh²(x) = 1

Applying this to our equation:

d/dx[tanh(x)] = 1/cosh²(x)

We can express this in terms of the hyperbolic secant function:

d/dx[tanh(x)] = sech²(x)

Thus, derivative of tanh(x) is sech²(x)

3. Solve the equation: cosh(x) = 2

Given, cosh(x) = 2

Recall that cosh(x) = ([Tex]e^x + e^{-x}[/Tex]) / 2

Substituting this into our equation:

(e^x + e^-x) / 2 = 2

Multiply bth sides by 2:

e^x + e^(-x) = 4

Let’s substitute y = e^x. This means e^(-x) = 1/y

y + 1/y = 4

Multiply all terms by y

y² + 1 = 4y

Rearrange to standard quadratic form

y² – 4y + 1 = 0

Now we can solve this using the quadratic formula: y = (-b ± √(b² – 4ac)) / 2a

Here, a = 1, b = -4, and c = 1

y = (4 ± √(16 – 4)) / 2 = (4 ± √12) / 2 = 2 ± √3

Remember that y = e^x, so:

e^x = 2 + √3 or e^x = 2 – √3

Taking the natural logarithm of both sides:

x = ln(2 + √3) or x = ln(2 – √3)

However, since cosh(x) is always greater than or equal to 1, we only keep the positive solution.

Therefore, the solution is x = ln(2 + √3) ≈ 1.3169578…

4. Prove that sinh(x+y) = sinh(x)cosh(y) + cosh(x)sinh(y)

We know that: sinh(t) = (e^t – e^(-t)) / 2

Let’s apply this to sinh(x+y):

sinh(x+y) = (e^(x+y) – e^-(x+y)) / 2

Expand the exponents:

sinh(x+y) = (e^x × e^y – e^(-x) × e^(-y)) / 2

Now, let’s add and subtract (e^x × e^(-y) + e^(-x) × e^y) / 2:

sinh(x+y) = (e^x × e^y – e^(-x) × e^(-y) + e^x × e^(-y) – e^x × e^(-y) + e^(-x) × e^y – e^(-x) × e^y) / 2

Group terms:

sinh(x+y) = () / 2

Factor out common terms:

sinh(x+y) = (e^x × (e^y – e^(-y)) / 2) + ((e^x – e^(-x)) / 2) × (e^y + e^(-y)) / 2

Recognize the definitions of sinh and cosh:

sinh(x+y) = sinh(y) × cosh(x) + sinh(x) × cosh(y)

Thus, we have proved the identity.

5. Calculate the value of sinh(ln(2)).

We know that: sinh(x) = (e^x – e^(-x)) / 2

Let’s substitute ln(2) for x:

sinh(ln(2)) = (e^ln(2) – e^(-ln(2))) / 2

Simplify the exponentials:

e^ln(2) = 2

e^(-ln(2)) = 1/2

Substitute these values:

sinh(ln(2)) = (2 – 1/2) / 2

Simplify:

sinh(ln(2)) = 3/4 = 0.75

Therefore, value of sinh(ln(2)) is 0.75.

6. Find the inverse of y = tanh(x).

Given, y = tanh(x)

Recall that tanh(x) = (e^x – e^(-x)) / (e^x + e^(-x))

Substitute this into our equation

y = (e^x – e^(-x)) / (e^x + e^(-x))

Cross multiply

y(e^x + e^(-x)) = e^x – e^(-x)

Expand

ye^x + ye^(-x) = e^x – e^(-x)

Group terms

e^x – ye^x = -e^(-x) – ye^(-x)

Factor

e^x(1 – y) = e^(-x)(-1 – y)

Divide both sides by -1 – y

e^x × ((y – 1) / (y + 1)) = e^(-x)

Take the natural log of both sides

x + ln((y – 1) / (y + 1)) = -x

Solve for x

2x = ln((1 + y) / (1 – y))

Divide by 2

x = (1/2) × ln((1 + y) / (1 – y))

Therefore, inverse of tanh(x) is arctanh(y) = (1/2) × ln((1 + y) / (1 – y))

7. Prove that cosh²(x) – sinh²(x) = 1

We start with,

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

sinh(x) = (e^x – e^(-x))/2

Square both

cosh²(x) = ((e^x + e^(-x))/2)²

sinh²(x) = ((e^x – e^(-x))/2)²

Expand

cosh²(x) = (e^(2x) + 2 + e^(-2x))/4

sinh²(x) = (e^(2x) – 2 + e^(-2x))/4

Subtract

cosh²(x) – sinh²(x) = ((e(2x) + 2 + e(-2x)) – (e(2x) – 2 + e(-2x))) / 4

Simplify:

cosh²(x) – sinh²(x) = 4 / 4 = 1

Thus, we have proved the identity.

8. Solve the equation: sinh(x) = 3

We start with sinh(x) = 3

Recall that sinh(x) = (e^x – e^(-x))/2

Substitute this into our equation

(e^x – e^(-x)) / 2 = 3

Multiply both sides by 2

e^x – e^(-x) = 6

Let y = e^x. Then e^(-x) = 1/y

Substituting

y – 1/y = 6

Multiply by y

y² – 1 = 6y

Rearrange

y² – 6y – 1 = 0

This is a quadratic. We’ll solve using the quadratic formula: y = (-b ± √(b² – 4ac)) / 2a

Here, a = 1, b = -6, and c = -1

y = (6 ± √(36 + 4)) / 2 = (6 ± √40) / 2 = 3 ± √10

Since y = e^x, we have:

e^x = 3 + √10

Taking the natural log of both sides:

x = ln(3 + √10) ≈ 1.8184464…

9. Find the derivative of cosh²(3x).

Let u = 3x, so we can use the chain rule

d/dx[cosh²(3x)] = d/du[cosh²(u)] × du/dx

First, let’s find d/du[cosh²(u)]

Using the power rule and the derivative of cosh(u)

d/du[cosh²(u)] = 2cosh(u) × sinh(u)

We know that du/dx = 3

Applying the chain rule

d/dx[cosh²(3x)] = 2cosh(3x) × sinh(3x) × 3

Simplify

d/dx[cosh²(3x)] = 6cosh(3x)sinh(3x)

10. Evaluate the integral: ∫sinh(x) dx

We know that the derivative of cosh(x) is sinh(x)

Therefore, the integral of sinh(x) is cosh(x) plus a constant of integration:

∫sinh(x)dx = cosh(x) + C

Hyperbolic Functions- FAQs

What are Fundamental Hyperbolic Identities?

• cosh²(x) – sinh²(x) = 1
• tanh² (x) + sech² (x) = 1

What is the Derivative of tanh(x)?

The derivative of tanh(x) is sech²(x).

Are Hyperbolic Functions Periodic?

No, unlike trigonometric functions, hyperbolic functions are not periodic.

What is the Range of cosh(x)?

The range of cosh(x) is all real numbers greater than or equal to 1.

How do you Calculate sinh(x)?

Value of sinh(x) is calculated using the formula, sinh(x) = {e^x – e^-x}/ 2.