Hyperbolic Relations — Hard Practice Quiz

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

Formulas & key concepts

Sine to Hyperbolic Sine

$$\sin(ix) = i\sinh x$$

Where: \(i\) = imaginary unit

Cosine to Hyperbolic Cosine

$$\cos(ix) = \cosh x$$

Where: \(i\) = imaginary unit

Tangent to Hyperbolic Tangent

$$\tan(ix) = i\tanh x$$

Where: \(i\) = imaginary unit

Practice quiz

  1. What is the simplified expression for $\sin(ix - y)$ in terms of hyperbolic and trigonometric functions, where $x$ and $y$ are real numbers?

    • $i\sinh x \cos y - \cosh x \sin y$
    • $\sinh x \cos y - i\cosh x \sin y$
    • $i\cosh x \sin y - \sinh x \cos y$
    • $\cosh x \sin y - i\sinh x \cos y$

    Answer: $i\sinh x \cos y - \cosh x \sin y$

  2. If $x$ is a real number, what is the value of $\cos^2(ix) + \sin^2(ix)$?

    • $1$
    • $-1$
    • $\cosh(2x)$
    • $\sinh(2x)$

    Answer: $1$

  3. If $\tan(ix) = 2i$, what is the value of $\cosh(2x)$?

    • $-\frac{5}{3}$
    • $\frac{5}{3}$
    • $\frac{3}{5}$
    • $-\frac{3}{5}$

    Answer: $-\frac{5}{3}$

  4. Given the identity $\tan(ix) = i\tanh x$, if $x$ is a non-zero real number, which of the following statements is true about $\tan(ix)$?

    • It is always a purely real number.
    • It is always a purely imaginary number.
    • It is always a complex number with both real and imaginary parts.
    • Its magnitude is always greater than $1$.

    Answer: It is always a purely imaginary number.

  5. If $\sin(ix) = 3i$ and $\cos(iy) = 2$, what is $\tan(i(x+y))$?

    • $i \frac{6+\sqrt{30}}{2\sqrt{10}+3\sqrt{3}}$
    • $\frac{6+\sqrt{30}}{2\sqrt{10}+3\sqrt{3}}$
    • $i \frac{6-\sqrt{30}}{2\sqrt{10}-3\sqrt{3}}$
    • $\frac{6-\sqrt{30}}{2\sqrt{10}-3\sqrt{3}}$

    Answer: $i \frac{6+\sqrt{30}}{2\sqrt{10}+3\sqrt{3}}$

  6. Express $\cot(ix)$ in terms of hyperbolic functions.

    • $-i\coth x$
    • $i\coth x$
    • $-i\tanh x$
    • $i\tanh x$

    Answer: $-i\coth x$

  7. If $\cos(ix) - \sin(ix) = \frac{5}{4} - i\frac{3}{4}$, what is the value of $e^{-x}$?

    • $\frac{1}{2}$
    • $2$
    • $\frac{5}{4}$
    • $\frac{3}{4}$

    Answer: $\frac{1}{2}$

  8. If $\tan(ix) = A$ and $\tan(iy) = B$, what is $\tan(i(x+y))$ in terms of $A$ and $B$?

    • $\frac{A+B}{1-AB}$
    • $\frac{A+B}{1+AB}$
    • $\frac{A-B}{1-AB}$
    • $\frac{A-B}{1+AB}$

    Answer: $\frac{A+B}{1-AB}$

  9. If $\sin(ix) = A$ and $\cos(ix) = B$, what is $\tan(2ix)$ in terms of $A$ and $B$?

    • $\frac{2AB}{B^2-A^2}$
    • $\frac{2AB}{A^2-B^2}$
    • $\frac{A^2-B^2}{2AB}$
    • $\frac{B^2-A^2}{2AB}$

    Answer: $\frac{2AB}{B^2-A^2}$

  10. Express $\sin(x+iy)$ in terms of real trigonometric and hyperbolic functions.

    • $\sin x \cosh y + i\cos x \sinh y$
    • $\cos x \cosh y + i\sin x \sinh y$
    • $\sin x \sinh y + i\cos x \cosh y$
    • $\cos x \sinh y + i\sin x \cosh y$

    Answer: $\sin x \cosh y + i\cos x \sinh y$

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