Hyperbolic Relations — Practice Quiz

A Trigonometry cheat sheet for Hyperbolic Relations — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Practice quiz

1. What is the equivalent hyperbolic function for $\sin(i\theta)$?

   • $i\sinh \theta$
   • $\cosh \theta$
   • $i\tanh \theta$
   • $\sinh \theta$

   Answer: $i\sinh \theta$

2. Express $\cos(i2x)$ in terms of a hyperbolic function.

   • $\cosh(2x)$
   • $i\sinh(2x)$
   • $i\cosh(2x)$
   • $\sinh(2x)$

   Answer: $\cosh(2x)$

3. If $f(x) = \tan(ix)$, what is $f(x)$ in terms of hyperbolic functions?

   • $i\tanh x$
   • $\tanh x$
   • $i\coth x$
   • $\text{sech } x$

   Answer: $i\tanh x$

4. Given $\cosh(3y)$, which of the following is equivalent?

   • $\cos(i3y)$
   • $\sin(i3y)$
   • $i\sinh(3y)$
   • $\tan(i3y)$

   Answer: $\cos(i3y)$

5. Simplify the expression $i\sinh(x/2)$.

   • $\sin(ix/2)$
   • $\cos(ix/2)$
   • $\tan(ix/2)$
   • $\cosh(x/2)$

   Answer: $\sin(ix/2)$

6. Which of the following is equal to $i\tanh(5t)$?

   • $\tan(i5t)$
   • $\sin(i5t)$
   • $\cos(i5t)$
   • $\coth(i5t)$

   Answer: $\tan(i5t)$

7. Evaluate $\cos(i\pi)$.

   • $\cosh(\pi)$
   • $i\sinh(\pi)$
   • $1$
   • $-1$

   Answer: $\cosh(\pi)$

8. If $z = ix$, what is $\sin(z) + \cos(z)$?

   • $i\sinh x + \cosh x$
   • $\sinh x + \cosh x$
   • $i\sinh x - \cosh x$
   • $\sinh x - i\cosh x$

   Answer: $i\sinh x + \cosh x$

9. What is the imaginary part of $\tan(i\frac{\pi}{4})$?

   • $\tanh(\frac{\pi}{4})$
   • $0$
   • $i\tanh(\frac{\pi}{4})$
   • $1$

   Answer: $\tanh(\frac{\pi}{4})$

10. Simplify $\sin(-ix)$.

    • $-i\sinh x$
    • $i\sinh x$
    • $\cosh x$
    • $-\cosh x$

    Answer: $-i\sinh x$