Hyperbolic Tangent Of Double Value
The hyperbolic tangent of twice a given variable can be found according to the following formula for all real numbers x :
tanh 2 x = (2 tanh x )/(1 + tanh ^{2} x )
Half Values
Hyperbolic Sine Of Half Value
The hyperbolic sine of half any given variable can be found according to the following formula for all nonnegative real numbers x :
sinh ( x /2) = [(1 – cosh x )/2] ^{1/2}
For negative real numbers x , the formula is:
sinh ( x /2) = –[(1 – cosh x )/2] ^{1/2}
Hyperbolic Cosine Of Half Value
The hyperbolic cosine of half any given variable can be found according to the following formula for all real numbers x :
cosh ( x /2) = [(1 + cosh x )/2] ^{1/2}
Sums and Differences
Hyperbolic Sine Of Sum
The hyperbolic sine of the sum of two variables x and y can be found according to the following formula for all real numbers x and y :
sinh ( x + y ) = sinh x cosh y + cosh x sinh y
Hyperbolic Cosine Of Sum
The hyperbolic cosine of the sum of two variables x and y can be found according to the following formula for all real numbers x and y:
cosh ( x + y ) = cosh x cosh y + sinh x sinh y
Hyperbolic Tangent Of Sum
The hyperbolic tangent of the sum of two variables x and y can be found according to the following formula for all real numbers x and y:
tanh ( x + y ) = (tanh x + tanh y )/(1 + tanh x tanh y )
Hyperbolic Sine Of Difference
The hyperbolic sine of the difference between two variables x and y can be found according to the following formula for all real numbers x and y:
sinh ( x – y ) = sinh x cosh y – cosh x sinh y
Hyperbolic Cosine Of Difference
The hyperbolic cosine of the difference between two variables x and y can be found according to the following formula for all real numbers x and y:
cosh ( x – y ) = cosh x cosh y – sinh x sinh y
Hyperbolic Tangent Of Difference
The hyperbolic tangent of the difference between two variables x and y can be found according to the following formula for all real numbers x and y , provided the product of tanh x and tanh y is not equal to 1:
tanh ( x – y ) = (tanh x – tanh y )/(1 – tanh x tanh y )
Hyperbolic Function Facts Practice Problems
Practice 1
Based on the above formulas, find a formula for the hyperbolic sine of three times a given value. That is, find a general formula for sinh 3 x . Express the answer in terms of functions of x only.
Solution 1
Let’s start out by supposing that y = 2 x , so x + y = x + 2x = 3x . We have a formula for the hyperbolic sine of the sum of two values. It is:
sinh ( x + y ) = sinh x cosh y + cosh x sinh y
Substituting 2x in place of y , we know this:
sinh 3x = sinh ( x + 2 x ) = sinh x cosh 2 x + cosh x sinh 2 x
We have formulas to determine cosh 2 x and sinh 2 x . They are:
cosh 2 x = cosh ^{2} x + sinh ^{2} x
sinh 2 x = 2 sinh x cosh x
We can substitute these equivalents in the previous formula, getting this:
There are two other ways this problem can be solved, because there are three different formulas for the hyperbolic cosine of a double value.
Practice 2
Verify (approximately) the following formula for x = 3 and y = 2:
sinh ( x = y ) = sinh x cosh y – cosh x sinh y
Solution 2
Let’s plug in the numbers:
 sing a calculator, we find these values based on the exponential formulas for the hyperbolic sine and cosine:
sinh 1 = 1.1752
sinh 2 = 3.6269
sinh 3 = 10.0179
cosh 2 = 3.7622
cosh 3 = 10.0677
We can put these values into the second formula above and see if the numbers add up. We should find that the following expression calculates out to approximately sinh 1, or 1.1752. Here we go:
This is close enough, considering that error accumulation occurs when performing repeated calculations with numbers that aren’t exact. Error accumulation involves the idiosyncrasies of scientific notation and significant figures. When significant figures aren’t taken seriously, they (or their lack) can cause trouble for experimental scientists, engineers, surveyors, and navigators.
Practice problems for these concepts can be found at: Hyperbolic Functions Practice Test
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