Answer
Verified
394.2k+ 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
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Assertion CNG is a better fuel than petrol Reason It class 11 chemistry CBSE
How does pressure exerted by solid and a fluid differ class 8 physics CBSE
Number of valence electrons in Chlorine ion are a 16 class 11 chemistry CBSE
What are agricultural practices? Define
What does CNG stand for and why is it considered to class 10 chemistry CBSE
The rate of evaporation depends on a Surface area b class 9 chemistry CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
State whether the following statement is true or false class 11 physics CBSE
A night bird owl can see very well in the night but class 12 physics CBSE