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Popular Trigonometry >

tanh(0.5)

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Solution

tanh(0.5)

Solution

e2−1e2−2e21​e​+1​
+1
Decimal
0.46211…
Solution steps
tanh(0.5)
=tanh(21​)
Rewrite using trig identities:e2−1e2−2e21​e​+1​
tanh(21​)
Use the Hyperbolic identity: tanh(x)=ex+e−xex−e−x​=e21​+e−21​e21​−e−21​​
e21​+e−21​e21​−e−21​​=e2−1e2−2e21​e​+1​
e21​+e−21​e21​−e−21​​
Apply exponent rule: a−b=ab1​e−21​=e​1​=e21​+e​1​e21​−e​1​​
Join e21​+e​1​:e​e21​e​+1​
e21​+e​1​
Convert element to fraction: e21​=e​e21​e​​=e​e21​e​​+e​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=e​e21​e​+1​
=e​e21​e​+1​e21​−e​1​​
Join e21​−e​1​:e​e21​e​−1​
e21​−e​1​
Convert element to fraction: e21​=e​e21​e​​=e​e21​e​​−e​1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=e​e21​e​−1​
=e​e21​e​+1​e​e21​e​−1​​
Divide fractions: dc​ba​​=b⋅ca⋅d​=e​(e21​e​+1)(e21​e​−1)e​​
Cancel the common factor: e​=e21​e​+1e21​e​−1​
Rationalize e21​e​+1e21​e​−1​:e2−1e2−2e21​e​+1​
e21​e​+1e21​e​−1​
Multiply by the conjugate e21​e​−1e21​e​−1​=(e21​e​+1)(e21​e​−1)(e21​e​−1)(e21​e​−1)​
(e21​e​−1)(e21​e​−1)=e2−2e21​e​+1
(e21​e​−1)(e21​e​−1)
Apply exponent rule: ab⋅ac=ab+c(e21​e​−1)(e21​e​−1)=(e21​e​−1)1+1=(e21​e​−1)1+1
Add the numbers: 1+1=2=(e21​e​−1)2
Apply Perfect Square Formula: (a−b)2=a2−2ab+b2a=e21​e​,b=1
=(e21​e​)2−2e21​e​⋅1+12
Simplify (e21​e​)2−2e21​e​⋅1+12:e2−2e21​e​+1
(e21​e​)2−2e21​e​⋅1+12
Apply rule 1a=112=1=(e21​e​)2−2⋅1⋅e21​e​+1
(e21​e​)2=e2
(e21​e​)2
Apply exponent rule: (a⋅b)n=anbn=(e​)2(e21​)2
(e21​)2:e
Apply exponent rule: (ab)c=abc=e21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=e
=e(e​)2
(e​)2:e
Apply radical rule: a​=a21​=(e21​)2
Apply exponent rule: (ab)c=abc=e21​⋅2
21​⋅2=1
21​⋅2
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=e
=ee
Apply exponent rule: ab⋅ac=ab+cee=e1+1=e1+1
Add the numbers: 1+1=2=e2
2e21​e​⋅1=2e21​e​
2e21​e​⋅1
Multiply the numbers: 2⋅1=2=2e21​e​
=e2−2e21​e​+1
=e2−2e21​e​+1
(e21​e​+1)(e21​e​−1)=e2−1
(e21​e​+1)(e21​e​−1)
e21​e​=e
e21​e​
Apply exponent rule: ab⋅ac=ab+ce21​e​=e21​e21​=e21​+21​=e21​+21​
Add similar elements: 21​+21​=2⋅21​=e2⋅21​
Multiply 2⋅21​:1
2⋅21​
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=e1
Apply rule a1=a=e
=(e+1)(e21​e​−1)
e21​e​=e
e21​e​
Apply exponent rule: ab⋅ac=ab+ce21​e​=e21​e21​=e21​+21​=e21​+21​
Add similar elements: 21​+21​=2⋅21​=e2⋅21​
Multiply 2⋅21​:1
2⋅21​
Multiply fractions: a⋅cb​=ca⋅b​=21⋅2​
Cancel the common factor: 2=1
=e1
Apply rule a1=a=e
=(e+1)(e−1)
Apply Difference of Two Squares Formula: (a+b)(a−b)=a2−b2a=e,b=1=e2−12
Apply rule 1a=112=1=e2−1
=e2−1e2−2e21​e​+1​
=e2−1e2−2e21​e​+1​
=e2−1e2−2e21​e​+1​
=e2−1e2−2e21​e​+1​

Popular Examples

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Frequently Asked Questions (FAQ)

  • What is the value of tanh(0.5) ?

    The value of tanh(0.5) is (e^2-2e^{1/2}sqrt(e)+1)/(e^2-1)
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