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hyperbolic functions
A LevelFurther Mathematics~5 min read
Overview
This lesson introduces hyperbolic functions, which are analogous to trigonometric functions but defined using the hyperbola rather than the circle. We will explore their definitions, fundamental identities, and basic calculus properties, essential for advanced mathematical applications.
Introduction to Hyperbolic Functions and Their Definitions
Hyperbolic functions are a set of functions that share many properties with the more familiar trigonometric functions, but are defined using the exponential function. They arise naturally in many areas of mathematics and physics, particularly in the study of catenaries (the shape of a hanging chain)...
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Key Concepts
- Hyperbolic Sine (sinh x): Defined as (e^x - e^-x) / 2
- Hyperbolic Cosine (cosh x): Defined as (e^x + e^-x) / 2
- Hyperbolic Tangent (tanh x): Defined as sinh x / cosh x = (e^x - e^-x) / (e^x + e^-x)
- Fundamental Identity: cosh^2 x - sinh^2 x = 1, analogous to sin^2 x + cos^2 x = 1
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Exam Tips
- →Memorise the exponential definitions of sinh x and cosh x, as all identities and derivatives can be derived from these.
- →Be able to prove the fundamental identity cosh^2 x - sinh^2 x = 1, and use it to derive other identities like 1 - tanh^2 x = sech^2 x.
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