What are the Key Formulas and Properties of Hyperbolic Functions?
The concept of hyperbolic functions plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. These functions serve as the counterparts of trigonometric functions but are defined using hyperbolas rather than circles. Understanding hyperbolic functions is crucial for students in Class 11 and 12, as well as those preparing for JEE, NEET, and other competitive exams.
What Is Hyperbolic Functions?
Hyperbolic functions are mathematical functions analogous to trigonometric functions, but they are based on the properties of the unit hyperbola instead of the unit circle. The main hyperbolic functions are sinh, cosh, tanh, and their reciprocals and inverses. You’ll find this concept applied in areas such as calculus, differential equations, and complex numbers.
Key Formula for Hyperbolic Functions
Here are the standard formulas for the six basic hyperbolic functions, defined using exponential functions:
| Function | Definition | Domain | Range |
|---|---|---|---|
| sinh x | \( \frac{e^x - e^{-x}}{2} \) | \( \mathbb{R} \) | \( \mathbb{R} \) |
| cosh x | \( \frac{e^x + e^{-x}}{2} \) | \( \mathbb{R} \) | \( [1, \infty) \) |
| tanh x | \( \frac{sinh\,x}{cosh\,x} = \frac{e^x - e^{-x}}{e^x + e^{-x}} \) | \( \mathbb{R} \) | \( (-1, 1) \) |
| coth x | \( \frac{cosh\,x}{sinh\,x} \) | \( \mathbb{R}, x \neq 0 \) | \( (-\infty, -1)\cup(1, \infty) \) |
| sech x | \( \frac{1}{cosh\,x} \) | \( \mathbb{R} \) | \( (0, 1] \) |
| csch x | \( \frac{1}{sinh\,x} \) | \( \mathbb{R}, x \neq 0 \) | \( (-\infty, 0)\cup(0, \infty) \) |
Cross-Disciplinary Usage
Hyperbolic functions are not only useful in Maths but also play an important role in Physics, Computer Science, and engineering. For instance, they are used in the study of special relativity, architecture (like the shape of suspension bridge cables), and signal processing. Students preparing for competitive exams like JEE or NEET will see their relevance in calculus and applied problems.
Properties and Identities of Hyperbolic Functions
- sinh(−x) = −sinh(x) (odd function)
- cosh(−x) = cosh(x) (even function)
- cosh2x − sinh2x = 1
- tanh x = sinh x / cosh x
- sum and difference formulas:
sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
Step-by-Step Illustration
- Express \( e^x \) and \( e^{-x} \) in terms of sinh x and cosh x:
Add = cosh x + sinh x = \( \frac{e^x + e^{-x}}{2} \) + \( \frac{e^x - e^{-x}}{2} \) = \( e^x \)
Subtract = cosh x − sinh x = \( e^{-x} \) - Solve: Simplify \( \frac{sinh\,x}{cosh\,x} \):
\( \frac{(e^x - e^{-x})/2}{(e^x + e^{-x})/2} = \frac{e^x - e^{-x}}{e^x + e^{-x}} = tanh\,x \)
Derivatives and Integrals of Hyperbolic Functions
| Function | Derivative | Integral |
|---|---|---|
| sinh x | cosh x | cosh x + C |
| cosh x | sinh x | sinh x + C |
| tanh x | sech2x | ln|cosh x| + C |
Inverse Hyperbolic Functions
The inverses of hyperbolic functions are also important. They are called area hyperbolic functions and are used to find the value of x for a given value of a hyperbolic function.
| Function | Inverse Formula |
|---|---|
| sinh-1x | ln(x + √(x2 + 1)) |
| cosh-1x | ln(x + √(x2 - 1)) |
| tanh-1x | 0.5 × ln((1 + x)/(1 − x)) |
Speed Trick or Vedic Shortcut
Here’s a quick way to remember the derivatives of hyperbolic functions: They are similar to trigonometric functions, but there is no sign change. For example, the derivative of sinh x is cosh x, and the derivative of cosh x is sinh x (not negative). This trick helps avoid common mistakes in differentiation during board or JEE exams.
Example Trick: If you know the derivatives of sine and cosine (sin' = cos, cos' = -sin), just drop the minus sign for their hyperbolic counterparts!
Tricks like this can save you precious seconds in competitive exams. Vedantu’s live classes cover these subtle differences for better exam performance.
Try These Yourself
- Find the value of sinh(0) and cosh(0).
- Simplify \( tanh^2 x + sech^2 x \).
- Prove: cosh2x – sinh2x = 1.
- Differentiate tanh x with respect to x.
Frequent Errors and Misunderstandings
- Confusing hyperbolic functions with trigonometric functions—especially in derivatives or signs.
- Forgetting domains in inverse hyperbolic formulas.
- Mixing up properties: e.g., thinking sinh x is always positive like sin x (it's not—it takes all real values).
Relation to Other Concepts
The idea of hyperbolic functions connects closely with topics such as Trigonometric Functions and Exponential Functions. Mastering this helps with understanding advanced calculus, differential equations, and the transformation of equations from the circular to hyperbolic form. Check out Derivatives of Parametric Functions for more applications.
Applications and Problem-Solving
Hyperbolic functions appear in real-world problems, such as the shape of cables on suspension bridges (called catenary curves) and in the solutions of certain integrals. Here’s a classic example:
1. Model the cable of a suspension bridge as \( y(x) = a \cosh(x/a) + b \).2. If the lowest point of the cable is at (0, 30) and the cable is attached to pillars at x = ±140, y = 80:
3. Substitute into the formula:
\( 30 = a \cosh(0/a) + b \implies 30 = a + b \)
\( 80 = a \cosh(140/a) + b \)
4. Solve these equations to find values for a and b using algebra (or a calculator).
5. Use these values to write the final equation for the bridge cable.
Summary Table & Quick Revision
| Formula | Key Points |
|---|---|
| cosh2x − sinh2x = 1 | Pythagorean-like identity |
| sinh(−x) = −sinh(x), cosh(−x) = cosh(x) | Even/Odd properties |
| tanh x = sinh x / cosh x | Definition from basics |
| sinh-1x = ln(x + √(x2 + 1)) | Inverse formula |
Classroom Tip
A quick way to remember hyperbolic identities is: They mostly look like trigonometric identities—but with signs changed here and there. Vedantu’s teachers often use visual comparisons and color-coded charts to make learning these more memorable during live classes.
We explored hyperbolic functions—from definition, formulas, examples, common mistakes, and their connection to other mathematical topics. Continue practicing with Vedantu to become confident in solving problems using this concept. For deeper insights, visit Integration by Substitution and Inverse Trigonometric Functions for related techniques.
FAQs on Hyperbolic Functions in Maths: Formulas, Properties & Applications
1. What are hyperbolic functions in Maths and how are they defined?
Hyperbolic functions are analogues of the standard trigonometric functions, but they are defined using a hyperbola rather than a circle. The three primary hyperbolic functions are defined using the exponential function 'e':
Hyperbolic Sine (sinh x) = (ex - e-x)/2
Hyperbolic Cosine (cosh x) = (ex + e-x)/2
Hyperbolic Tangent (tanh x) = sinh(x)/cosh(x) = (ex - e-x)/(ex + e-x)
2. What are the six main hyperbolic functions?
The six main hyperbolic functions are direct counterparts to the six trigonometric functions. They include the three primary functions and their reciprocals:
Hyperbolic Sine (sinh x)
Hyperbolic Cosine (cosh x)
Hyperbolic Tangent (tanh x)
Hyperbolic Cosecant (csch x) = 1/sinh x
Hyperbolic Secant (sech x) = 1/cosh x
Hyperbolic Cotangent (coth x) = 1/tanh x
3. How do hyperbolic functions fundamentally differ from trigonometric functions?
The main differences lie in their geometric basis, periodicity, and definitions:
Geometric Basis: Trigonometric functions are based on the unit circle (x²+y²=1), while hyperbolic functions are based on the unit hyperbola (x²-y²=1).
Periodicity: Trigonometric functions like sin(x) and cos(x) are periodic. Hyperbolic functions (for real number inputs) are not periodic.
Definition: Hyperbolic functions are defined using the exponential function (e^x), whereas trigonometric functions are defined by angles within a circle.
4. What is a key real-world example of a hyperbolic function?
A classic real-world application is the shape of a flexible cable or chain hanging freely between two points, which forms a curve called a catenary. This shape is precisely described by the hyperbolic cosine (cosh x) function. This principle is used in designing suspension bridges and understanding the shape of power lines.
5. Why does the main hyperbolic identity use subtraction (cosh²x - sinh²x = 1)?
This identity arises directly from the exponential definitions of sinh(x) and cosh(x). When you substitute their formulas into the expression cosh²x - sinh²x and simplify, the exponential terms cancel out, leaving exactly 1. It is the direct analogue of the Pythagorean identity cos²θ + sin²θ = 1, which is derived from the geometry of a circle, just as the hyperbolic identity is derived from the geometry of a hyperbola.
6. What do the graphs of the basic hyperbolic functions look like?
Each primary hyperbolic function has a distinct graph:
y = sinh(x): An odd function that passes through the origin (0,0) and increases from -∞ to +∞, resembling a steep cubic curve.
y = cosh(x): An even function that forms a 'catenary' curve. Its lowest point is (0,1), and it is always positive.
y = tanh(x): An S-shaped curve that passes through the origin and is bounded between two horizontal asymptotes at y = -1 and y = 1.
7. Are hyperbolic functions periodic like sine and cosine?
No, for real numbers, the basic hyperbolic functions like sinh(x), cosh(x), and tanh(x) are not periodic. Unlike trigonometric functions which repeat their values over a fixed interval (e.g., 2π), the values of hyperbolic functions like sinh(x) and cosh(x) grow indefinitely as x increases and do not repeat.
8. How do the derivatives of hyperbolic functions compare to trigonometric ones?
The derivatives are very similar, but with a key difference in signs that students must note:
The derivative of sinh(x) is cosh(x).
The derivative of cosh(x) is sinh(x). (Note: there is no negative sign, unlike the derivative of cos(x), which is -sin(x)).
The derivative of tanh(x) is sech²(x).
9. What is the importance of inverse hyperbolic functions?
Inverse hyperbolic functions, such as sinh⁻¹(x) and cosh⁻¹(x), are important because they can be expressed using natural logarithms. For example, `sinh⁻¹(x) = ln(x + √(x²+1))`. This connection provides a powerful method for solving complex integrals in calculus that would otherwise be very difficult.
