Overview of Hyperbolic Functions in JEE Maths

Hyperbolic functions are an important topic in JEE Maths, forming an extension of trigonometry to hyperbolas. They are analogous to the circular trigonometric functions but are defined using exponential functions. At StudentBro.in, we provide a comprehensive guide covering definitions, properties, formulas, identities, derivatives, applications, and practice problems to help students excel in JEE Main & Advanced exams.

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Definition of Hyperbolic Functions

Hyperbolic functions are based on exponential functions and defined as:

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

tanh x = sinh x / cosh x, coth x = cosh x / sinh x

sech x = 1 / cosh x, csch x = 1 / sinh x

These functions are widely used in mathematical modeling, engineering, and JEE problem-solving.

Basic Properties of Hyperbolic Functions

• Even-Odd Properties: sinh(−x) = −sinh x (odd), cosh(−x) = cosh x (even), tanh(−x) = −tanh x (odd)
• Fundamental Identity: cosh² x − sinh² x = 1
• Reciprocal Identities: coth x = 1 / tanh x, sech x = 1 / cosh x, csch x = 1 / sinh x

Addition and 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)

Derivatives of Hyperbolic Functions

• d/dx(sinh x) = cosh x
• d/dx(cosh x) = sinh x
• d/dx(tanh x) = sech² x
• d/dx(coth x) = −csch² x
• d/dx(sech x) = −sech x tanh x
• d/dx(csch x) = −csch x coth x

Integration Formulas

• ∫ sinh x dx = cosh x + C
• ∫ cosh x dx = sinh x + C
• ∫ tanh x dx = ln|cosh x| + C
• ∫ coth x dx = ln|sinh x| + C
• ∫ sech x dx = tan⁻¹(sinh x) + C
• ∫ csch x dx = ln|tanh(x/2)| + C

Hyperbolic Equations

Common hyperbolic equations appear in exponentially related problems:

sinh x = k, cosh x = k, tanh x = k

Solve using inverse hyperbolic functions:

x = sinh⁻¹ k = ln(k + √(k² + 1))

x = cosh⁻¹ k = ln(k + √(k² − 1))

x = tanh⁻¹ k = 1/2 ln((1 + k) / (1 − k))

Applications in JEE Maths

• Algebra and Calculus: Solve equations involving exponential functions
• Trigonometry: Solve hyperbolic analogues of trigonometric problems
• Coordinate Geometry: Solve curves related to hyperbolas
• Physics: Model problems in optics, mechanics, and waves

Tips for JEE Exam Preparation

• Memorize all basic identities and addition formulas
• Practice derivatives and integrals of hyperbolic functions
• Apply inverse hyperbolic formulas carefully in equations
• Solve previous years’ JEE questions for speed and accuracy
• Draw hyperbolic curves for visual understanding

StudentBro.in Hyperbolic Functions Section

• Step-by-step explanations for sinh, cosh, tanh, coth, sech, csch
• Worked examples for derivatives, integrals, and identities
• Practice questions with solutions and shortcuts
• Tips for quick problem-solving in JEE exams
• Revision notes for last-minute preparation

Practice Questions

1. Solve sinh x = 2
2. Evaluate ∫ cosh x dx
3. Prove cosh² x − sinh² x = 1
4. Find derivative of tanh x
5. Solve cosh x = 5 using inverse hyperbolic function

Conclusion

Hyperbolic functions are a vital topic in JEE Maths, closely related to trigonometry and exponential functions. Understanding identities, derivatives, integrals, and inverse functions allows students to tackle a wide range of problems efficiently.

At StudentBro.in, we provide a complete guide from basics to advanced problem-solving, making hyperbolic functions easy, practical, and exam-oriented. Mastery of this topic ensures students can confidently solve all hyperbolic function problems in JEE Main & Advanced exams.