JEE Maths Notes: Chapter on Hyperbolic Functions

1. Introduction to Hyperbolic Functions

Hyperbolic functions are analogs of the trigonometric functions but for a hyperbola instead of a circle. These functions play an essential role in calculus and are particularly useful in solving problems involving differential equations, integrals, and in various engineering fields. The basic hyperbolic functions include sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), coth (hyperbolic cotangent), sec (hyperbolic secant), and csc (hyperbolic cosecant).

2. Hyperbolic Sine and Cosine

The two primary hyperbolic functions are the hyperbolic sine and hyperbolic cosine, which are defined as:

• Hyperbolic Sine (sinh x): sinh x = (e^x - e^(-x)) / 2
• Hyperbolic Cosine (cosh x): cosh x = (e^x + e^(-x)) / 2

These functions resemble the sine and cosine functions but are based on the exponential function.

3. Other Hyperbolic Functions

Other hyperbolic functions are derived from the basic sinh and cosh functions:

• Hyperbolic Tangent (tanh x): tanh x = sinh x / cosh x = (e^x - e^(-x)) / (e^x + e^(-x))
• Hyperbolic Cotangent (coth x): coth x = cosh x / sinh x = (e^x + e^(-x)) / (e^x - e^(-x))
• Hyperbolic Secant (sech x): sech x = 1 / cosh x = 2 / (e^x + e^(-x))
• Hyperbolic Cosecant (csch x): csch x = 1 / sinh x = 2 / (e^x - e^(-x))

These functions are used to express various relationships between hyperbolic sine and cosine, just as trigonometric functions relate sine and cosine.

4. Properties of Hyperbolic Functions

Similar to trigonometric functions, hyperbolic functions also satisfy various identities:

• Pythagorean Identity: cosh² x - sinh² x = 1
• Sum and Difference Identities:
  • sinh(x ± y) = sinh x cosh y ± cosh x sinh y
  • cosh(x ± y) = cosh x cosh y ± sinh x sinh y
• Double Angle Formulas:
  • sinh 2x = 2 sinh x cosh x
  • cosh 2x = cosh² x + sinh² x

These identities are fundamental for solving various hyperbolic equations and simplifying expressions involving hyperbolic functions.

5. Graphs of Hyperbolic Functions

The graphs of hyperbolic functions resemble those of trigonometric functions but differ in the nature of their curves. The graphs for sinh x and cosh x exhibit exponential growth or decay, reflecting their connection to exponential functions. The graph of cosh x is U-shaped, while sinh x is similar to an odd function with rapid growth in both positive and negative directions.

6. Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverse operations of the hyperbolic functions, similar to how inverse trigonometric functions operate in trigonometry. The inverse hyperbolic functions are used to solve equations involving hyperbolic functions. These functions include:

• Inverse Hyperbolic Sine (arsinh x): arsinh x = ln(x + √(x² + 1))
• Inverse Hyperbolic Cosine (arcosh x): arcosh x = ln(x + √(x² - 1))
• Inverse Hyperbolic Tangent (artanh x): artanh x = (1/2) ln((1 + x) / (1 - x))
• Inverse Hyperbolic Cotangent (arcoth x): arcoth x = (1/2) ln((x + 1) / (x - 1))
• Inverse Hyperbolic Secant (arsech x): arsech x = ln((1 + √(1 - x²)) / x)
• Inverse Hyperbolic Cosecant (arcsch x): arcsch x = ln((√(1 + x²) + 1) / x)

These inverse functions allow you to solve equations involving hyperbolic expressions.

7. Applications of Hyperbolic Functions

Hyperbolic functions are widely used in various fields:

• Engineering: Applied in the analysis of electrical circuits, wave equations, and systems involving exponential growth or decay.
• Physics: Used in the study of special relativity, hyperbolic geometry, and fluid mechanics.
• Mathematics: Used in solving differential equations, optimization problems, and in hyperbolic geometry.

8. Summary

Hyperbolic functions are crucial in advanced mathematics, and mastering them is essential for solving many types of problems in calculus, differential equations, and physics. The properties and applications of hyperbolic functions make them an indispensable tool in JEE preparation. Regular practice with hyperbolic identities, inverses, and applications ensures a thorough understanding of the topic, which is key for achieving success in JEE exams.