Hyperbolic Functions
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Questions and Answers

If cosec θ = cosh z, then the value of sinh z is

  • tan θ
  • cot θ (correct)
  • sin θ
  • sec θ
  • What is the value of tanh(x + y) when cosh x = sec θ and sec y = cosec θ?

  • tan θ (correct)
  • tanh θ
  • coth θ
  • cot θ
  • If cosh z = 3, then what is the value of sinh z?

  • √2
  • √5 (correct)
  • 2√3
  • 2√2
  • What is the value of coth x when cosec θ = cosh x?

    <p>cot θ</p> Signup and view all the answers

    If tanh x = 1/2, then what is the value of cosh x?

    <p>2/√3</p> Signup and view all the answers

    If log sec x = y, then the value of y is

    <p>2 coth-1 (cosec 2 - 1)</p> Signup and view all the answers

    The value of cosh-1 (sec x) is

    <p>log(1 + sin x / cos x)</p> Signup and view all the answers

    If tanh x = 1/2, then the value of coth x is

    <p>3</p> Signup and view all the answers

    The value of sinh-1 (tan x) is

    <p>log(sec x + tan x)</p> Signup and view all the answers

    If cosech x = 2, then the value of x is

    <p>log(2 + √(2^2 - 1))</p> Signup and view all the answers

    If $f(x) = \cosh{x} + \sinh{x}$, then $f(p) = f(x) \cdot f(y)$, what is the value of $p$?

    <p>f(x_1) \cdot f(x_2) \cdot... \cdot f(x_n)</p> Signup and view all the answers

    What is the value of $\cosh^2{x} - \sinh^2{x}$?

    <p>1</p> Signup and view all the answers

    If $\sin{x} \cosh{y} = \cos{\theta}$ and $\cos{x} \sinh{y} = \sin{\theta}$, what is the value of $\sinh^2{y}$?

    <p>1 - cos(2x)</p> Signup and view all the answers

    What is the value of $\tanh{x} + \tanh{y}$?

    <p>1 + \tanh{x} \tanh{y}</p> Signup and view all the answers

    What is the value of $\cosh{2x} - 1$?

    <p>2 \sinh^2{x}</p> Signup and view all the answers

    What is the formula for tan⁻¹(cos θ + i sin θ) when cos θ > 0?

    <p>sin⁻¹(sin θ) + i log(sin θ + 1 + sin θ)</p> Signup and view all the answers

    If x = log(y + √(y² + 1)), then what is y in terms of x?

    <p>sinh x</p> Signup and view all the answers

    If cosh⁻¹ x = log(2 + 3), then what is the value of x?

    <p>2</p> Signup and view all the answers

    Which of the following is true about inverse hyperbolic functions?

    <p>They are many-valued.</p> Signup and view all the answers

    What is the formula for sinh⁻¹(cos θ + i sin θ)?

    <p>cos⁻¹(sin θ) - i log(sin θ + 1 + sin θ)</p> Signup and view all the answers

    What is the value of sin$^{-1}$(cosec $\theta$) in terms of $\theta$?

    <p>$\pi/2 + i$ log(cot $\theta$)</p> Signup and view all the answers

    What is the real part of cos$^{-1}$(cos $\theta + i$ sin $\theta$)?

    <p>$\pi/2$</p> Signup and view all the answers

    What is the value of sin$^{-1}$(sin $\theta - i$ cos $\theta$)?

    <p>$\pi - \theta - i$ log(cos $\theta$)</p> Signup and view all the answers

    What is the value of cos$^{-1}$(cos $\theta + i$ sin $\theta$) in terms of $\theta$?

    <p>$\theta + i$ log(sin $\theta$)</p> Signup and view all the answers

    What is the domain of the hyperbolic sine function?

    <p>R</p> Signup and view all the answers

    If $e^x - e^{-x} = 4$, what is the value of $ ext{sinh} x$?

    <p>2</p> Signup and view all the answers

    What is the range of the hyperbolic cosine function?

    <p>[1, ∞)</p> Signup and view all the answers

    If $ ext{cosh} x = 2$, what is the value of $ ext{sinh} x$?

    <p>√3</p> Signup and view all the answers

    What is the value of $ ext{tanh} 0$?

    <p>0</p> Signup and view all the answers

    What is the relationship between sin(x) and sinh(ix)?

    <p>sin(x) = -i sinh(ix)</p> Signup and view all the answers

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