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Hyperbolic Trig Functions Greg Kelly, Hanford High School, Richland, Washington

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Consider the following two functions: These functions show up frequently enough that they have been given names.

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The behavior of these functions shows such remarkable parallels to trig functions, that they have been given similar names.

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Hyperbolic Sine: (pronounced “cinch x”) Hyperbolic Cosine: (pronounced “kosh x”)

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Hyperbolic Tangent: “tansh (x)” Hyperbolic Cotangent: “cotansh (x)” Hyperbolic Secant: “sech (x)” Hyperbolic Cosecant: “cosech (x)”

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First, an easy one: Now, if we have “trig-like” functions, it follows that we will have “trig-like” identities.

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(This one doesn’t really have an analogy in trig.)

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Note that this is similar to but not the same as: Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.

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Derivatives can be found relatively easily using the definitions. Surprise, this is positive!

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(quotient rule)

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All of the derivatives are similar to trig functions except for some of the signs. Sinh, Cosh and Tanh are positive. The others are negative

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Integral formulas can be written from the derivative formulas. On the TI-89, the hyperbolic functions are under: 2ndMATHHyperbolic Or you can use the catalog.

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Teacher find the derivative of (e x + e -x )/2 Student: Oh that’s a cinch

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