Answer
397.2k+ views
Hint: Now, in this question hyperbolic trigonometric functions of $ \cos $ and $ \sin $ are mentioned. The hyperbolic functions are to be written in the exponential functions. Thereafter we will have to simplify the obtained equation to find the trigonometric equation’s value.
Formula used: We will have to use the formula of hyperbolic cos function, $ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $ and hyperbolic sine function, $ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $ .
Complete step-by-step answer:
According to the given information, we have
The first function is $ A = $ $ \sinh (3) $
The formula for $ \sinh x $ in terms of exponential function is,
$ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
$ \sinh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ ...(1) $
The first function is $ B = $ $ \cosh (3) $
The formula for $ \cosh x $ in terms of exponential function is,
$ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
$ \cosh (3) $ $ = \dfrac{{{e^3} + {e^{ - 3}}}}{2} $ $ ...(2) $
According to the given data we have to calculate $ \sinh (3) $ $ - \cosh (3) $ , which is equal to,
$ \sinh (3) $ $ - \cosh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ - \dfrac{{{e^3} + {e^{ - 3}}}}{2} $
\[ \Rightarrow \dfrac{{{e^3} - {e^{ - 3}} - ({e^3} - {e^{ - 3}})}}{2}\]
\[ \Rightarrow \dfrac{{ - 2{e^{ - 3}}}}{2}\]\[ = - {e^{ - 3}}\]
So, the correct answer is “Option B”.
Note: In order to solve problems of this type the key is to have a basic understanding of trigonometric equations and values and also learn its implications. Hyperbolic functions are very similar to the trigonometric functions but are expressed in the form of exponential functions and the most common of them are $ \cosh x $ and $ \sinh x $ .
The formula for $ \cosh x $ in terms of exponential function is,
$ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
The formula for $ \sinh x $ in terms of exponential function is,
$ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $
Formula used: We will have to use the formula of hyperbolic cos function, $ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $ and hyperbolic sine function, $ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $ .
Complete step-by-step answer:
According to the given information, we have
The first function is $ A = $ $ \sinh (3) $
The formula for $ \sinh x $ in terms of exponential function is,
$ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
$ \sinh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ ...(1) $
The first function is $ B = $ $ \cosh (3) $
The formula for $ \cosh x $ in terms of exponential function is,
$ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
Substituting x=3 in the above given equation we get,
$ \cosh (3) $ $ = \dfrac{{{e^3} + {e^{ - 3}}}}{2} $ $ ...(2) $
According to the given data we have to calculate $ \sinh (3) $ $ - \cosh (3) $ , which is equal to,
$ \sinh (3) $ $ - \cosh (3) $ \[ = \dfrac{{{e^3} - {e^{ - 3}}}}{2}\] $ - \dfrac{{{e^3} + {e^{ - 3}}}}{2} $
\[ \Rightarrow \dfrac{{{e^3} - {e^{ - 3}} - ({e^3} - {e^{ - 3}})}}{2}\]
\[ \Rightarrow \dfrac{{ - 2{e^{ - 3}}}}{2}\]\[ = - {e^{ - 3}}\]
So, the correct answer is “Option B”.
Note: In order to solve problems of this type the key is to have a basic understanding of trigonometric equations and values and also learn its implications. Hyperbolic functions are very similar to the trigonometric functions but are expressed in the form of exponential functions and the most common of them are $ \cosh x $ and $ \sinh x $ .
The formula for $ \cosh x $ in terms of exponential function is,
$ \cosh x $ $ = \dfrac{{{e^x} + {e^{ - x}}}}{2} $
The formula for $ \sinh x $ in terms of exponential function is,
$ \sinh x $ $ = \dfrac{{{e^x} - {e^{ - x}}}}{2} $
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)