Problem 1
Question
Find the numerical value of each expression. (a) \(\sinh 0\) (b) \(\cosh 0\)
Step-by-Step Solution
Verified Answer
\(\sinh 0 = 0\) and \(\cosh 0 = 1\).
1Step 1: Understanding Hyperbolic Functions
The hyperbolic sine function, denoted as \(\sinh x\), is defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\). The hyperbolic cosine function, denoted as \(\cosh x\), is defined as \(\cosh x = \frac{e^x + e^{-x}}{2}\). We use these definitions to find the expressions at zero.
2Step 2: Calculating \(\sinh 0\)
Substitute \(x = 0\) into the hyperbolic sine function: \(\sinh 0 = \frac{e^0 - e^{-0}}{2} = \frac{1 - 1}{2} = 0\). Therefore, \(\sinh 0 = 0\).
3Step 3: Calculating \(\cosh 0\)
Substitute \(x = 0\) into the hyperbolic cosine function: \(\cosh 0 = \frac{e^0 + e^{-0}}{2} = \frac{1 + 1}{2} = 1\). Therefore, \(\cosh 0 = 1\).
Key Concepts
sinh functioncosh functionexponential functions
sinh function
The sinh function is short for the hyperbolic sine function. It is similar to the trigonometric sine function but relates to hyperbolas instead of circles. It is defined using the exponential functions as \[\sinh x = \frac{e^x - e^{-x}}{2}.\] This formula captures the difference between two exponential expressions divided by 2. A key feature of the \(\sinh\) function is its odd symmetry, meaning \(\sinh(-x) = -\sinh x\).
When you calculate \(\sinh 0\), you substitute 0 into the formula, which results in:\[\sinh 0 = \frac{e^0 - e^{-0}}{2} = \frac{1 - 1}{2} = 0.\]This demonstrates that at zero, the hyperbolic sine function equals zero. This value reflects the function passing through the origin of the hyperbolic curve.
When you calculate \(\sinh 0\), you substitute 0 into the formula, which results in:\[\sinh 0 = \frac{e^0 - e^{-0}}{2} = \frac{1 - 1}{2} = 0.\]This demonstrates that at zero, the hyperbolic sine function equals zero. This value reflects the function passing through the origin of the hyperbolic curve.
- Useful in mathematical models involving hyperbolas.
- Has applications in engineering and physics when modeling hyperbolic paths.
cosh function
The cosh function, or hyperbolic cosine function, is another important hyperbolic function. It resembles the trigonometric cosine function but applies to hyperbolas. It's defined as:\[\cosh x = \frac{e^x + e^{-x}}{2}.\]This formula shows an average of two exponential expressions. Unlike \(\sinh x\), this function is even, which means \(\cosh(-x) = \cosh x\).
To find \(\cosh 0\), you substitute 0 into its formula:\[\cosh 0 = \frac{e^0 + e^{-0}}{2} = \frac{1 + 1}{2} = 1.\]This calculation shows that at zero, the hyperbolic cosine function equals one. At this point, the function represents the lowest point on the hyperbolic curve.
To find \(\cosh 0\), you substitute 0 into its formula:\[\cosh 0 = \frac{e^0 + e^{-0}}{2} = \frac{1 + 1}{2} = 1.\]This calculation shows that at zero, the hyperbolic cosine function equals one. At this point, the function represents the lowest point on the hyperbolic curve.
- Certain real-world applications include calculations for electricity and optics.
- It is widely used in the analysis of catenary shapes in construction and architecture.
exponential functions
Exponential functions are fundamental in mathematics with a constant base and a variable exponent. The expression \(e^x\) represents exponential growth, while \(e^{-x}\) depicts exponential decay. These functions are crucial for calculating both the \(\sinh\) and \(\cosh\) functions.
The natural number \(e\) is approximately equal to 2.718, and it is a key component of these functions due to its unique properties in calculus and analysis. In the context of hyperbolic functions, the interplay between \(e^x\) and \(e^{-x}\) allows for a balance between these two exponential modes.
Exponential functions are not only theoretical but have a range of practical applications:
The natural number \(e\) is approximately equal to 2.718, and it is a key component of these functions due to its unique properties in calculus and analysis. In the context of hyperbolic functions, the interplay between \(e^x\) and \(e^{-x}\) allows for a balance between these two exponential modes.
Exponential functions are not only theoretical but have a range of practical applications:
- Modeling population growth, radioactive decay, and financial interest.
- Act as a foundation for understanding natural phenomena in various fields of science and engineering.
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