1.7E: Exercises
- Page ID
- 116877
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Evaluations
- Evaluate the given function. Round your answers to 3 decimal places, if necessary.
- \( \coth{(-7)} \)
- \( \operatorname{csch}{(\ln{(3)})} \)
- \( \cosh{(3)} \)
- \( \sinh{(0)} \)
- \( \tanh{(1)} \)
- \( \operatorname{sech}{(\ln{(7)})} \)
- \( \sinh^{-1}{(3)} \)
- \( \cosh^{-1}{\left(\frac{3}{2}\right)} \)
- \( \tanh^{-1}{\left(\frac{1}{2}\right)} \)
Simplifications
- Rewrite the following expressions in terms of exponentials and simplify.
- \(2\cosh(\ln x)\)
- \(\cosh 4x+\sinh 4x\)
- \(\cosh 2x−\sinh 2x\)
- \(\ln(\cosh x+\sinh x)+\ln(\cosh x−\sinh x)\)
Proofs
- Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \cosh{(x + y)} = \cosh{(x)} \cosh{(y)} + \sinh{(x)} \sinh{(y)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \sinh{(x + y)} = \sinh{(x)} \cosh{(y)} + \cosh{(x)} \sinh{(y)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \cosh{(2x)} = \cosh^2{(x)} + \sinh^2{(x)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \sinh{(2x)} = 2 \sinh{(x)} \cosh{(x)} \nonumber \] - Prove that \( f(x) = \cosh{(x)} \) is an even function.
- Prove that \( f(x) = \sinh{(x)} \) is an odd function.
- Prove the identitiy. Your work should be legible, and all your logic should be clear and justified.
\[\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)\nonumber\] - 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?
- 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?