Answer
Verified
423.9k+ views
Hint:
Arc functions are also termed as arcus functions, anti-trigonometric functions, inverse trigonometric functions or cyclometric functions. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.
There are particularly six inverse trig functions for each trigonometric ratio. The inverse of six important trigonometric functions are:
Arcsine
Arccosine
Arctangent
Arc cotangent
Arc secant
Arc cosecant
Arc of some trigonometric function is the same as the inverse of the function. Arc prefix is commonly used to name the trigonometric functions. For example, \[\arcsin x\] = \[\dfrac{1}{2}\]means the angle whose sine is making \[\dfrac{1}{2}\]. Thus, we can say that \[{{\sin }^{-1}}(x)\]= \[\dfrac{1}{2}\] or x = \[{{30}^{\circ }}\].
To solve this question, find the range of all three trigonometric functions and add them to find the right answer.
Complete step by step solution:
Step 1: Let us first find the range of those three trigonometric functions separately.
Since, we know \[\arcsin x+\arccos x\] = \[\dfrac{\pi }{2}\]
Therefore, range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])
Step 2: Minimum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\] + (\[-\dfrac{\pi }{2}\]) = 0
Maximum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\]+\[\dfrac{\pi }{2}\] = \[\pi \]
Hence, the range of \[\arcsin x+\arccos x+\arctan x\] = (0, \[\pi \])
Note:
To solve this question, the students should be familiar with trigonometric functions. The basics of trigonometry values are a must. Try to visualise such questions through graphs that can perform such trigonometric functionalities. Once you get familiar with trigonometric graphs and functions you can easily find the values of inverse trigonometric functions. Also, for range, observe the round brackets () are used to write the range as the values 0 and \[\pi \]are not included since the range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])
Arc functions are also termed as arcus functions, anti-trigonometric functions, inverse trigonometric functions or cyclometric functions. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.
There are particularly six inverse trig functions for each trigonometric ratio. The inverse of six important trigonometric functions are:
Arcsine
Arccosine
Arctangent
Arc cotangent
Arc secant
Arc cosecant
Arc of some trigonometric function is the same as the inverse of the function. Arc prefix is commonly used to name the trigonometric functions. For example, \[\arcsin x\] = \[\dfrac{1}{2}\]means the angle whose sine is making \[\dfrac{1}{2}\]. Thus, we can say that \[{{\sin }^{-1}}(x)\]= \[\dfrac{1}{2}\] or x = \[{{30}^{\circ }}\].
To solve this question, find the range of all three trigonometric functions and add them to find the right answer.
Complete step by step solution:
Step 1: Let us first find the range of those three trigonometric functions separately.
Since, we know \[\arcsin x+\arccos x\] = \[\dfrac{\pi }{2}\]
Therefore, range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])
Step 2: Minimum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\] + (\[-\dfrac{\pi }{2}\]) = 0
Maximum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\]+\[\dfrac{\pi }{2}\] = \[\pi \]
Hence, the range of \[\arcsin x+\arccos x+\arctan x\] = (0, \[\pi \])
Note:
To solve this question, the students should be familiar with trigonometric functions. The basics of trigonometry values are a must. Try to visualise such questions through graphs that can perform such trigonometric functionalities. Once you get familiar with trigonometric graphs and functions you can easily find the values of inverse trigonometric functions. Also, for range, observe the round brackets () are used to write the range as the values 0 and \[\pi \]are not included since the range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE