Answer

Verified

412.2k+ 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.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail