Solved on Jan 17, 2024

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

STEP 1

Assumptions
1. The hyperbolic tangent function, denoted as tanh(x)\tanh(x), is defined as tanh(x)=sinh(x)cosh(x)\tanh(x) = \frac{\sinh(x)}{\cosh(x)} where sinh(x)\sinh(x) is the hyperbolic sine of xx and cosh(x)\cosh(x) is the hyperbolic cosine of xx.
2. The hyperbolic sine and cosine functions can be expressed in terms of exponential functions: sinh(x)=exex2\sinh(x) = \frac{e^x - e^{-x}}{2} and cosh(x)=ex+ex2\cosh(x) = \frac{e^x + e^{-x}}{2}.
3. The value of tanh(0)\tanh(0) and tanh(1)\tanh(1) can be found by plugging in the values of xx into the definition of tanh(x)\tanh(x).
4. The answers should be rounded to five decimal places.

STEP 2

Calculate 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)}

STEP 3

Compute sinh(0)\sinh(0) and cosh(0)\cosh(0) using the exponential expressions for hyperbolic sine and cosine.
sinh(0)=e0e02\sinh(0) = \frac{e^0 - e^{-0}}{2} cosh(0)=e0+e02\cosh(0) = \frac{e^0 + e^{-0}}{2}

STEP 4

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

STEP 5

Substitute the values from STEP_4 into the expressions for sinh(0)\sinh(0) and cosh(0)\cosh(0).
sinh(0)=112=0\sinh(0) = \frac{1 - 1}{2} = 0 cosh(0)=1+12=1\cosh(0) = \frac{1 + 1}{2} = 1

STEP 6

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

STEP 7

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

STEP 8

Next, calculate 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)}

STEP 9

Compute sinh(1)\sinh(1) and cosh(1)\cosh(1) using the exponential expressions for hyperbolic sine and cosine.
sinh(1)=e1e12\sinh(1) = \frac{e^1 - e^{-1}}{2} cosh(1)=e1+e12\cosh(1) = \frac{e^1 + e^{-1}}{2}

STEP 10

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

STEP 11

Substitute the values from STEP_10 into the expressions for sinh(1)\sinh(1) and cosh(1)\cosh(1).
sinh(1)=e1e2\sinh(1) = \frac{e - \frac{1}{e}}{2} cosh(1)=e+1e2\cosh(1) = \frac{e + \frac{1}{e}}{2}

STEP 12

Calculate the values of sinh(1)\sinh(1) and cosh(1)\cosh(1).
sinh(1)=e1e2=e212e\sinh(1) = \frac{e - \frac{1}{e}}{2} = \frac{e^2 - 1}{2e} cosh(1)=e+1e2=e2+12e\cosh(1) = \frac{e + \frac{1}{e}}{2} = \frac{e^2 + 1}{2e}

STEP 13

Now, substitute the values of sinh(1)\sinh(1) and cosh(1)\cosh(1) into the definition of tanh(1)\tanh(1).
tanh(1)=e212ee2+12e\tanh(1) = \frac{\frac{e^2 - 1}{2e}}{\frac{e^2 + 1}{2e}}

STEP 14

Simplify the expression for tanh(1)\tanh(1) by canceling out common factors.
tanh(1)=e21e2+1\tanh(1) = \frac{e^2 - 1}{e^2 + 1}

STEP 15

Use a calculator to find the numerical value of tanh(1)\tanh(1).
tanh(1)e21e2+1\tanh(1) \approx \frac{e^2 - 1}{e^2 + 1}

STEP 16

Round the result to five decimal places.
tanh(1)0.76159\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\tanh(0) = 0.00000 (b) tanh(1)0.76159\tanh(1) \approx 0.76159

Was this helpful?
banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ContactInfluencer programPolicyTerms
TwitterInstagramFacebookTikTokDiscord