Write $9 \sinh (x)+4 \cosh (x)$ in terms of $e^x$ and $e^{-x}$.
$(1) e^x+(\quad) e^{-x}$
Answer (2)
Express sinh ( x ) and cosh ( x ) in terms of e x and e − x : sinh ( x ) = 2 e x − e − x and cosh ( x ) = 2 e x + e − x .
Substitute these expressions into 9 sinh ( x ) + 4 cosh ( x ) .
Simplify the resulting expression to get 2 13 e x − 2 5 e − x .
Identify the coefficient of e − x , which is − 2 5 .
Explanation
Problem Analysis We are asked to express 9 sinh ( x ) + 4 cosh ( x ) in terms of e x and e − x . We know the definitions of sinh ( x ) and cosh ( x ) in terms of exponentials.
Definitions of Hyperbolic Functions Recall the definitions: sinh ( x ) = 2 e x − e − x cosh ( x ) = 2 e x + e − x
Substitution Substitute these definitions into the given expression: 9 sinh ( x ) + 4 cosh ( x ) = 9 ( 2 e x − e − x ) + 4 ( 2 e x + e − x ) = 2 9 e x − 9 e − x + 2 4 e x + 4 e − x
Simplification Combine like terms: 2 9 e x − 9 e − x + 2 4 e x + 4 e − x = 2 9 e x + 4 e x − 9 e − x + 4 e − x = 2 13 e x − 5 e − x = 2 13 e x − 2 5 e − x
Final Expression Thus, we have 9 sinh ( x ) + 4 cosh ( x ) = 2 13 e x − 2 5 e − x . We want to write this in the form ( 1 ) e x + ( ) e − x . Therefore, the expression is 2 13 e x + ( − 2 5 ) e − x
Final Answer The coefficient of e x is 2 13 and the coefficient of e − x is − 2 5 . Therefore, the answer is − 2 5 .
Examples
Understanding hyperbolic functions and their exponential forms is crucial in various fields like physics and engineering. For instance, the shape of a hanging cable (like a power line) is described by a hyperbolic cosine function (cosh). By expressing these functions in terms of exponentials, engineers can more easily analyze the forces acting on the cable and ensure its stability. Similarly, in special relativity, hyperbolic functions are used to describe the relationship between space and time coordinates, and their exponential forms simplify calculations involving Lorentz transformations.
To express 9 sinh ( x ) + 4 cosh ( x ) in terms of e x and e − x , we substitute the definitions of hyperbolic functions and simplify to get 2 13 e x − 2 5 e − x . Therefore, the coefficient for e − x is − 2 5 .
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