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1.6E: Exercises

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    Reading Questions

    1. What is the algebraic relation that defines the unit circle? What analogous relation defines the unit hyperbola (right branch)?
    2. How are \(\cosh(t)\) and \(\sinh(t)\) defined in terms of exponential functions?
    3. What is the Fundamental Hyperbolic Identity, and how does it relate to the Pythagorean identity for trigonometric functions?
    4. Express \(\tanh(t)\) in terms of \(\sinh(t)\) and \(\cosh(t)\), and then in terms of exponential functions.
    5. Which hyperbolic function represents the shape of a catenary (a hanging cable)?
    6. How does the graph of \(\cosh(x)\) differ from \(\sinh(x)\) in terms of symmetry and y-intercept?
    7. What are the horizontal asymptotes of \(y=\tanh(x)\)?
    8. Are all hyperbolic functions one-to-one? If not, which ones require domain restrictions to define their inverses, and what is a common restriction?
    9. How is \(\text{sinh}^{-1}x\) defined in terms of a natural logarithm?
    10. How is \(\text{cosh}^{-1}x\) defined in terms of a natural logarithm? What is the domain for this inverse function?
    11. Is \(\cosh(x)\) an even or odd function? How about \(\sinh(x)\)?

    Homework

    Evaluations

    1. Evaluate the given function. Round your answers to 3 decimal places, if necessary.
      1. \( \coth{(-7)} \)
      2. \( \operatorname{csch}{(\ln{(3)})} \)
      3. \( \cosh{(3)} \)
      4. \( \sinh{(0)} \)
      5. \( \tanh{(1)} \)
      6. \( \operatorname{sech}{(\ln{(7)})} \)
      7. \( \sinh^{-1}{(3)} \)
      8. \( \cosh^{-1}{\left(\frac{3}{2}\right)} \)
      9. \( \tanh^{-1}{\left(\frac{1}{2}\right)} \)

    Simplifications

    1. Rewrite the following expressions in terms of exponentials and simplify.
      1. \(2\cosh(\ln x)\)
      2. \(\cosh 4x+\sinh 4x\)
      3. \(\cosh 2x−\sinh 2x\)
      4. \(\ln(\cosh x+\sinh x)+\ln(\cosh x−\sinh x)\)

    Proofs

    1. Prove the identity. Your work should be legible, and all your logic should be clear and justified.
      \[ \cosh{(x + y)} = \cosh{(x)} \cosh{(y)} + \sinh{(x)} \sinh{(y)} \nonumber \]
    2. Prove the identity. Your work should be legible, and all your logic should be clear and justified.
      \[ \sinh{(x + y)} = \sinh{(x)} \cosh{(y)} + \cosh{(x)} \sinh{(y)} \nonumber \]
    3. Prove the identity. Your work should be legible, and all your logic should be clear and justified.
      \[ \cosh{(2x)} = \cosh^2{(x)} + \sinh^2{(x)} \nonumber \]
    4. Prove the identity. Your work should be legible, and all your logic should be clear and justified.
      \[ \sinh{(2x)} = 2 \sinh{(x)} \cosh{(x)} \nonumber \]
    5. Prove that \( f(x) = \cosh{(x)} \) is an even function.
    6. Prove that \( f(x) = \sinh{(x)} \) is an odd function.
    7. Prove the identitiy. Your work should be legible, and all your logic should be clear and justified.
      \[\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)\nonumber\]
    8. Prove the expression for \(\sinh^{−1}(x)\). Multiply \(x=\sinh(y)=\dfrac{e^y−e^{−y}}{2}\) by \(2e^y\) and solve for \(y\). Does your expression match the textbook?
    9. Prove the expression for \(\cosh^{−1}(x)\). Multiply \(x=\cosh(y)=\dfrac{e^y+e^{−y}}{2}\) by \(2e^y\) and solve for \(y\). Does your expression match the textbook?

    This page titled 1.6E: Exercises is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Roy Simpson.

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