What is Sinh on Calculator

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Reviewed by: **Dr. Anya Sharma**, Applied Mathematics Expert.

This calculator accurately computes the Hyperbolic Sine ($\sinh$) of any given real number $x$, using the standard exponential definition.

What is sinh on calculator

Calculated Hyperbolic Sine ($\sinh(x)$):

what is sinh on calculator Formula:

$$\sinh(x) = \frac{e^x – e^{-x}}{2}$$

Formula Sources: Wolfram MathWorld | Wikipedia Hyperbolic Function

Variables:

The calculator requires one main input variable for the hyperbolic sine calculation:

  • Angle/Value (x): The real number for which you want to calculate the hyperbolic sine. In many applications, this value is unitless, though it represents a concept related to area in hyperbolic geometry.

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What is the Hyperbolic Sine ($\sinh$)?

The Hyperbolic Sine function, denoted as $\sinh(x)$, is one of the fundamental hyperbolic functions. It is conceptually analogous to the familiar trigonometric sine function ($\sin(x)$), but its definition and geometric interpretation are based on the hyperbola $x^2 – y^2 = 1$ rather than the unit circle. Hyperbolic functions are crucial in solving linear differential equations, especially those concerning wave propagation, heat conduction, and the shape of a hanging cable (catenary).

Defined via Euler’s number ($e$), $\sinh(x)$ is an odd function, meaning $\sinh(-x) = -\sinh(x)$, and it passes through the origin $(0, 0)$. Unlike the periodic nature of $\sin(x)$, $\sinh(x)$ increases without limit as $|x|$ increases, exhibiting exponential growth.

How to Calculate $\sinh(x)$ (Example):

Let’s find the value of $\sinh(3)$ using the formula:

  1. Identify $x$: The input value is $x = 3$.
  2. Apply the Formula: The definition is $\sinh(x) = \frac{e^x – e^{-x}}{2}$.
  3. Calculate $e^x$ and $e^{-x}$:
    • $e^3 \approx 20.085537$
    • $e^{-3} \approx 0.049787$
  4. Find the Difference: $e^3 – e^{-3} \approx 20.085537 – 0.049787 = 19.035750$.
  5. Divide by Two: $\sinh(3) \approx \frac{19.035750}{2} \approx 10.017875$.

Frequently Asked Questions (FAQ):

  • What is the derivative of $\sinh(x)$?

    The derivative of the hyperbolic sine function is the hyperbolic cosine function: $\frac{d}{dx}\sinh(x) = \cosh(x)$.

  • What is the relationship between $\sinh(x)$ and $\sin(x)$?

    The two functions are related through the imaginary unit $i$: $\sinh(x) = -i \sin(ix)$. They are two parts of the same family of exponential functions.

  • Does $\sinh(x)$ have a maximum or minimum value?

    No. The range of $\sinh(x)$ is all real numbers ($-\infty$ to $+\infty$). It is a strictly increasing function, having no local maxima or minima.

  • What is the common Taylor series expansion for $\sinh(x)$?

    The Taylor series for $\sinh(x)$ centered at $x=0$ contains only odd powers of $x$: $$\sinh(x) = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \dots = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!}$$

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