Answer
424.2k+ views
Hint: To solve this question we will take the help of graph of functions $ y={{\sin }^{-1}}(\operatorname{sinx}) $ and $ y={{\cos }^{-1}}(cosx) $ . What we will do is we will check on which line the value 10 on graph $ y={{\sin }^{-1}}(\operatorname{sinx}) $ and $ y={{\cos }^{-1}}(cosx) $ as graphs of functions $ y={{\sin }^{-1}}(\operatorname{sinx}) $ and $ y={{\cos }^{-1}}(cosx) $ are continuous.
Then, finally we will evaluate the value of y and x and hence find y – x .
Complete step-by-step answer:
Before we solve the question to get value of y – x ,
Let us see the properties and graph of function $ y={{\sin }^{-1}}(\operatorname{sinx}) $ .
$ {{\sin }^{-1}}(\operatorname{sinx})=-\pi -x;\dfrac{-3\pi }{2}\le x\le \dfrac{-\pi }{2} $ ,
$ =x;\dfrac{-\pi }{2}\le x\le \dfrac{\pi }{2} $ ,
$ =\pi ;\dfrac{\pi }{2}\le x\le \dfrac{3\pi }{2} $ .
And, graph of $ y={{\sin }^{-1}}(\operatorname{sinx}) $ is given as ,
Let us see the properties and graph of the function $ y={{\cos }^{-1}}(cosx) $ .
$ {{\cos }^{-1}}(cosx)=-x;-\pi \le x\le 0 $
$ =x;0\le x\le \pi $
$ =2\pi -x;\pi \le x\le 2\pi $
And, graph of $ y={{\cos }^{-1}}(cosx) $ is given as ,
Now, let us solve for $ x={{\sin }^{-1}}(\sin 10) $ first,
As we see that, $ \dfrac{5\pi }{2}\le 10\le \dfrac{7\pi }{2} $
So, we can say that 10 lies on line $ 3\pi -x $ , so 10 will satisfy the equation $ y=3\pi -x $
Putting x = 10 in $ y=3\pi -x $ , we get
$ y=3\pi -10 $ .
Or, $ 3\pi -10={{\sin }^{-1}}(\sin 10) $ …… ( i )
Now, let us solve for $ y={{\cos }^{-1}}(cos10) $ ,
As we see that, $ 3\pi \le 10\le 4\pi $ ,
So, we can say that 10 lies on line $ 4\pi -x $ , so 10 will satisfy the equation $ y=4\pi -x $
Putting x = 10 in $ y=4\pi -x $ , we get
$ y=4\pi -10 $ .
Or, $ 4\pi -10={{\cos }^{-1}}(cos10) $ …..( ii )
Now, we have to find the value of, y – x that is $ {{\cos }^{-1}}(cos10)-{{\sin }^{-1}}(\sin 10) $ ,which is equals to
$ y-x=(4\pi -10)-(3\pi -10) $ .
Note: Graph of $ y={{\sin }^{-1}}(\operatorname{sinx}) $ and $ y={{\cos }^{-1}}(cosx) $ are very important function and graphs should be remembered while solving questions based on inverse trigonometric functions. The value of input should be checked on which line of the function does it lie carefully as it may change the output of the function.
Then, finally we will evaluate the value of y and x and hence find y – x .
Complete step-by-step answer:
Before we solve the question to get value of y – x ,
Let us see the properties and graph of function $ y={{\sin }^{-1}}(\operatorname{sinx}) $ .
$ {{\sin }^{-1}}(\operatorname{sinx})=-\pi -x;\dfrac{-3\pi }{2}\le x\le \dfrac{-\pi }{2} $ ,
$ =x;\dfrac{-\pi }{2}\le x\le \dfrac{\pi }{2} $ ,
$ =\pi ;\dfrac{\pi }{2}\le x\le \dfrac{3\pi }{2} $ .
And, graph of $ y={{\sin }^{-1}}(\operatorname{sinx}) $ is given as ,
![seo images](https://www.vedantu.com/question-sets/f9b10391-61d9-42f9-96b5-7a4eaaceebf61113773089602796497.png)
Let us see the properties and graph of the function $ y={{\cos }^{-1}}(cosx) $ .
$ {{\cos }^{-1}}(cosx)=-x;-\pi \le x\le 0 $
$ =x;0\le x\le \pi $
$ =2\pi -x;\pi \le x\le 2\pi $
And, graph of $ y={{\cos }^{-1}}(cosx) $ is given as ,
![seo images](https://www.vedantu.com/question-sets/41e1ccde-7b32-43ee-a7a9-ba077433e0933279466715986821339.png)
Now, let us solve for $ x={{\sin }^{-1}}(\sin 10) $ first,
As we see that, $ \dfrac{5\pi }{2}\le 10\le \dfrac{7\pi }{2} $
So, we can say that 10 lies on line $ 3\pi -x $ , so 10 will satisfy the equation $ y=3\pi -x $
Putting x = 10 in $ y=3\pi -x $ , we get
$ y=3\pi -10 $ .
Or, $ 3\pi -10={{\sin }^{-1}}(\sin 10) $ …… ( i )
Now, let us solve for $ y={{\cos }^{-1}}(cos10) $ ,
As we see that, $ 3\pi \le 10\le 4\pi $ ,
So, we can say that 10 lies on line $ 4\pi -x $ , so 10 will satisfy the equation $ y=4\pi -x $
Putting x = 10 in $ y=4\pi -x $ , we get
$ y=4\pi -10 $ .
Or, $ 4\pi -10={{\cos }^{-1}}(cos10) $ …..( ii )
Now, we have to find the value of, y – x that is $ {{\cos }^{-1}}(cos10)-{{\sin }^{-1}}(\sin 10) $ ,which is equals to
$ y-x=(4\pi -10)-(3\pi -10) $ .
Note: Graph of $ y={{\sin }^{-1}}(\operatorname{sinx}) $ and $ y={{\cos }^{-1}}(cosx) $ are very important function and graphs should be remembered while solving questions based on inverse trigonometric functions. The value of input should be checked on which line of the function does it lie carefully as it may change the output of the function.
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)