Solved on Jan 17, 2024

Find the numerical value of $\tanh(0)$ and $\tanh(1)$ , rounded to five decimal places.

STEP 1

Assumptions
1. The hyperbolic tangent function, denoted as $\tanh(x)$ , is defined as $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$ where $\sinh(x)$ is the hyperbolic sine of $x$ and $\cosh(x)$ is the hyperbolic cosine of $x$ .
2. The hyperbolic sine and cosine functions can be expressed in terms of exponential functions: $\sinh(x) = \frac{e^x - e^{-x}}{2}$ and $\cosh(x) = \frac{e^x + e^{-x}}{2}$ .
3. The value of $\tanh(0)$ and $\tanh(1)$ can be found by plugging in the values of $x$ into the definition of $\tanh(x)$ .
4. The answers should be rounded to five decimal places.

STEP 2

Calculate $\tanh(0)$ using the definition of the hyperbolic tangent function.
$\tanh(0) = \frac{\sinh(0)}{\cosh(0)}$

STEP 3

Compute $\sinh(0)$ and $\cosh(0)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(0) = \frac{e^0 - e^{-0}}{2}$ $\cosh(0) = \frac{e^0 + e^{-0}}{2}$

STEP 4

Evaluate the exponential functions at $x = 0$ .
$e^0 = 1$ $e^{-0} = 1$

STEP 5

Substitute the values from STEP_4 into the expressions for $\sinh(0)$ and $\cosh(0)$ .
$\sinh(0) = \frac{1 - 1}{2} = 0$ $\cosh(0) = \frac{1 + 1}{2} = 1$

STEP 6

Now, substitute the values of $\sinh(0)$ and $\cosh(0)$ into the definition of $\tanh(0)$ .
$\tanh(0) = \frac{0}{1} = 0$

STEP 7

The numerical value of $\tanh(0)$ , rounded to five decimal places, is:
$\tanh(0) = 0.00000$

STEP 8

Next, calculate $\tanh(1)$ using the definition of the hyperbolic tangent function.
$\tanh(1) = \frac{\sinh(1)}{\cosh(1)}$

STEP 9

Compute $\sinh(1)$ and $\cosh(1)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(1) = \frac{e^1 - e^{-1}}{2}$ $\cosh(1) = \frac{e^1 + e^{-1}}{2}$

STEP 10

Evaluate the exponential functions at $x = 1$ .
$e^1 = e$ $e^{-1} = \frac{1}{e}$

STEP 11

Substitute the values from STEP_10 into the expressions for $\sinh(1)$ and $\cosh(1)$ .
$\sinh(1) = \frac{e - \frac{1}{e}}{2}$ $\cosh(1) = \frac{e + \frac{1}{e}}{2}$

STEP 12

Calculate the values of $\sinh(1)$ and $\cosh(1)$ .
$\sinh(1) = \frac{e - \frac{1}{e}}{2} = \frac{e^2 - 1}{2e}$ $\cosh(1) = \frac{e + \frac{1}{e}}{2} = \frac{e^2 + 1}{2e}$

STEP 13

Now, substitute the values of $\sinh(1)$ and $\cosh(1)$ into the definition of $\tanh(1)$ .
$\tanh(1) = \frac{\frac{e^2 - 1}{2e}}{\frac{e^2 + 1}{2e}}$

STEP 14

Simplify the expression for $\tanh(1)$ by canceling out common factors.
$\tanh(1) = \frac{e^2 - 1}{e^2 + 1}$

STEP 15

Use a calculator to find the numerical value of $\tanh(1)$ .
$\tanh(1) \approx \frac{e^2 - 1}{e^2 + 1}$

STEP 16

Round the result to five decimal places.
$\tanh(1) \approx 0.76159$ (This is an approximate value; the exact value will depend on the calculator used.)
The numerical values of each expression are: (a) $\tanh(0) = 0.00000$ (b) $\tanh(1) \approx 0.76159$

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Find the numerical value of $\tanh(0)$ and $\tanh(1)$ , rounded to five decimal places.

STEP 1

STEP 2

Calculate $\tanh(0)$ using the definition of the hyperbolic tangent function.
$\tanh(0) = \frac{\sinh(0)}{\cosh(0)}$

STEP 3

Compute $\sinh(0)$ and $\cosh(0)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(0) = \frac{e^0 - e^{-0}}{2}$ $\cosh(0) = \frac{e^0 + e^{-0}}{2}$

STEP 4

Evaluate the exponential functions at $x = 0$ .
$e^0 = 1$ $e^{-0} = 1$

STEP 5

Substitute the values from STEP_4 into the expressions for $\sinh(0)$ and $\cosh(0)$ .
$\sinh(0) = \frac{1 - 1}{2} = 0$ $\cosh(0) = \frac{1 + 1}{2} = 1$

STEP 6

Now, substitute the values of $\sinh(0)$ and $\cosh(0)$ into the definition of $\tanh(0)$ .
$\tanh(0) = \frac{0}{1} = 0$

STEP 7

The numerical value of $\tanh(0)$ , rounded to five decimal places, is:
$\tanh(0) = 0.00000$

STEP 8

Next, calculate $\tanh(1)$ using the definition of the hyperbolic tangent function.
$\tanh(1) = \frac{\sinh(1)}{\cosh(1)}$

STEP 9

Compute $\sinh(1)$ and $\cosh(1)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(1) = \frac{e^1 - e^{-1}}{2}$ $\cosh(1) = \frac{e^1 + e^{-1}}{2}$

STEP 10

Evaluate the exponential functions at $x = 1$ .
$e^1 = e$ $e^{-1} = \frac{1}{e}$

STEP 11

Substitute the values from STEP_10 into the expressions for $\sinh(1)$ and $\cosh(1)$ .
$\sinh(1) = \frac{e - \frac{1}{e}}{2}$ $\cosh(1) = \frac{e + \frac{1}{e}}{2}$

STEP 12

Calculate the values of $\sinh(1)$ and $\cosh(1)$ .
$\sinh(1) = \frac{e - \frac{1}{e}}{2} = \frac{e^2 - 1}{2e}$ $\cosh(1) = \frac{e + \frac{1}{e}}{2} = \frac{e^2 + 1}{2e}$

STEP 13

Now, substitute the values of $\sinh(1)$ and $\cosh(1)$ into the definition of $\tanh(1)$ .
$\tanh(1) = \frac{\frac{e^2 - 1}{2e}}{\frac{e^2 + 1}{2e}}$

STEP 14

Simplify the expression for $\tanh(1)$ by canceling out common factors.
$\tanh(1) = \frac{e^2 - 1}{e^2 + 1}$

STEP 15

Use a calculator to find the numerical value of $\tanh(1)$ .
$\tanh(1) \approx \frac{e^2 - 1}{e^2 + 1}$

STEP 16

Round the result to five decimal places.
$\tanh(1) \approx 0.76159$ (This is an approximate value; the exact value will depend on the calculator used.)
The numerical values of each expression are: (a) $\tanh(0) = 0.00000$ (b) $\tanh(1) \approx 0.76159$

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Find the numerical value of tanh⁡(0)\tanh(0)tanh(0) and tanh⁡(1)\tanh(1)tanh(1), rounded to five decimal places.

STEP 1

STEP 2

Calculate tanh⁡(0)\tanh(0)tanh(0) using the definition of the hyperbolic tangent function.tanh⁡(0)=sinh⁡(0)cosh⁡(0)\tanh(0) = \frac{\sinh(0)}{\cosh(0)}tanh(0)=cosh(0)sinh(0)​

STEP 3

Compute sinh⁡(0)\sinh(0)sinh(0) and cosh⁡(0)\cosh(0)cosh(0) using the exponential expressions for hyperbolic sine and cosine.sinh⁡(0)=e0−e−02\sinh(0) = \frac{e^0 - e^{-0}}{2}sinh(0)=2e0−e−0​ cosh⁡(0)=e0+e−02\cosh(0) = \frac{e^0 + e^{-0}}{2}cosh(0)=2e0+e−0​

STEP 4

Evaluate the exponential functions at x=0x = 0x=0.e0=1e^0 = 1e0=1 e−0=1e^{-0} = 1e−0=1

STEP 5

Substitute the values from STEP_4 into the expressions for sinh⁡(0)\sinh(0)sinh(0) and cosh⁡(0)\cosh(0)cosh(0).sinh⁡(0)=1−12=0\sinh(0) = \frac{1 - 1}{2} = 0sinh(0)=21−1​=0 cosh⁡(0)=1+12=1\cosh(0) = \frac{1 + 1}{2} = 1cosh(0)=21+1​=1

STEP 6

Now, substitute the values of sinh⁡(0)\sinh(0)sinh(0) and cosh⁡(0)\cosh(0)cosh(0) into the definition of tanh⁡(0)\tanh(0)tanh(0).tanh⁡(0)=01=0\tanh(0) = \frac{0}{1} = 0tanh(0)=10​=0

STEP 7

The numerical value of tanh⁡(0)\tanh(0)tanh(0), rounded to five decimal places, is:tanh⁡(0)=0.00000\tanh(0) = 0.00000tanh(0)=0.00000

STEP 8

Next, calculate tanh⁡(1)\tanh(1)tanh(1) using the definition of the hyperbolic tangent function.tanh⁡(1)=sinh⁡(1)cosh⁡(1)\tanh(1) = \frac{\sinh(1)}{\cosh(1)}tanh(1)=cosh(1)sinh(1)​

STEP 9

Compute sinh⁡(1)\sinh(1)sinh(1) and cosh⁡(1)\cosh(1)cosh(1) using the exponential expressions for hyperbolic sine and cosine.sinh⁡(1)=e1−e−12\sinh(1) = \frac{e^1 - e^{-1}}{2}sinh(1)=2e1−e−1​ cosh⁡(1)=e1+e−12\cosh(1) = \frac{e^1 + e^{-1}}{2}cosh(1)=2e1+e−1​

STEP 10

Evaluate the exponential functions at x=1x = 1x=1.e1=ee^1 = ee1=e e−1=1ee^{-1} = \frac{1}{e}e−1=e1​

STEP 11

Substitute the values from STEP_10 into the expressions for sinh⁡(1)\sinh(1)sinh(1) and cosh⁡(1)\cosh(1)cosh(1).sinh⁡(1)=e−1e2\sinh(1) = \frac{e - \frac{1}{e}}{2}sinh(1)=2e−e1​​ cosh⁡(1)=e+1e2\cosh(1) = \frac{e + \frac{1}{e}}{2}cosh(1)=2e+e1​​

STEP 12

STEP 13

Now, substitute the values of sinh⁡(1)\sinh(1)sinh(1) and cosh⁡(1)\cosh(1)cosh(1) into the definition of tanh⁡(1)\tanh(1)tanh(1).tanh⁡(1)=e2−12ee2+12e\tanh(1) = \frac{\frac{e^2 - 1}{2e}}{\frac{e^2 + 1}{2e}}tanh(1)=2ee2+1​2ee2−1​​

STEP 14

Simplify the expression for tanh⁡(1)\tanh(1)tanh(1) by canceling out common factors.tanh⁡(1)=e2−1e2+1\tanh(1) = \frac{e^2 - 1}{e^2 + 1}tanh(1)=e2+1e2−1​

STEP 15

Use a calculator to find the numerical value of tanh⁡(1)\tanh(1)tanh(1).tanh⁡(1)≈e2−1e2+1\tanh(1) \approx \frac{e^2 - 1}{e^2 + 1}tanh(1)≈e2+1e2−1​

STEP 16

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Find the numerical value of $\tanh(0)$ and $\tanh(1)$ , rounded to five decimal places.

Calculate $\tanh(0)$ using the definition of the hyperbolic tangent function.
$\tanh(0) = \frac{\sinh(0)}{\cosh(0)}$

Compute $\sinh(0)$ and $\cosh(0)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(0) = \frac{e^0 - e^{-0}}{2}$ $\cosh(0) = \frac{e^0 + e^{-0}}{2}$

Evaluate the exponential functions at $x = 0$ .
$e^0 = 1$ $e^{-0} = 1$

Substitute the values from STEP_4 into the expressions for $\sinh(0)$ and $\cosh(0)$ .
$\sinh(0) = \frac{1 - 1}{2} = 0$ $\cosh(0) = \frac{1 + 1}{2} = 1$

Now, substitute the values of $\sinh(0)$ and $\cosh(0)$ into the definition of $\tanh(0)$ .
$\tanh(0) = \frac{0}{1} = 0$

The numerical value of $\tanh(0)$ , rounded to five decimal places, is:
$\tanh(0) = 0.00000$

Next, calculate $\tanh(1)$ using the definition of the hyperbolic tangent function.
$\tanh(1) = \frac{\sinh(1)}{\cosh(1)}$

Compute $\sinh(1)$ and $\cosh(1)$ using the exponential expressions for hyperbolic sine and cosine.
$\sinh(1) = \frac{e^1 - e^{-1}}{2}$ $\cosh(1) = \frac{e^1 + e^{-1}}{2}$

Evaluate the exponential functions at $x = 1$ .
$e^1 = e$ $e^{-1} = \frac{1}{e}$

Substitute the values from STEP_10 into the expressions for $\sinh(1)$ and $\cosh(1)$ .
$\sinh(1) = \frac{e - \frac{1}{e}}{2}$ $\cosh(1) = \frac{e + \frac{1}{e}}{2}$

Now, substitute the values of $\sinh(1)$ and $\cosh(1)$ into the definition of $\tanh(1)$ .
$\tanh(1) = \frac{\frac{e^2 - 1}{2e}}{\frac{e^2 + 1}{2e}}$

Simplify the expression for $\tanh(1)$ by canceling out common factors.
$\tanh(1) = \frac{e^2 - 1}{e^2 + 1}$

Use a calculator to find the numerical value of $\tanh(1)$ .
$\tanh(1) \approx \frac{e^2 - 1}{e^2 + 1}$