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Each of Exercises $1-4$ gives a value of sinh $x$ or cosh $x$ . Use the definitions and the identity cosh $^{2} x-\sinh ^{2} x=1$ to find the values of the remaining five hyperbolic functions.$$\sin x=-\frac{3}{4}$$

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Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Transcendental Functions

Section 7

Hyperbolic Functions

Functions

Derivatives

Differentiation

Differential Equations

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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Each of Exercises $1-4$ gi…

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in this example, we're going to be completing the table that's displayed here, where the one thing that's provided is that the hyperbolic sign of X is equal to negative 3/4. This gives us one freebie right away, since if we go too hyperbolic, Cosi can't we can just take the reciprocal that's found for sign, so we'll have negative 4/3 altogether. Next, let's work on the hyperbolic coastline function, and to do that will use the identity. That hyperbolic coastline squared minus hyperbolic sine squared is equal to one. If we substitute hyperbolic signed into this equation provided here will have hyperbolic co sign. A X minus 9/16 is equal to one. This implies that hyperbolic co sign of X is equal to 25 16th. If we take the square root of both sides of this equation, we get that hyperbolic co sign of X is equal to 5/4. We didn't bother with determining whether or not this is positive or negative, even though we did take the square root of an equation. The reason for that is hyperbolic. Co sign itself is always a positive quantity, so we get 5/4 right off the bat then, as we did before. Hyper Black Sea Kent is just the reciprocal, so we now have 4/5 and now we're ready to begin this row for tangent and hyperbolic co tangent. Where this should say hyper bulk co tangent. Adickes. Let's start with hyperbolic tangent where we'll have sign, which is negative. 3/4 Divide by Hyper Bowl co sign, which is 5/4 and this is negative 3/5 altogether. Then the last thing to do is to go do hyperbolic co tangent, which is just the reciprocal of the value to the left. So we have negative 5/3 here and the table is complete.

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