Domain and Range of Basic Inverse Trigonometric Functions
If f:[0,4π] →...
Question
If f:[0,4π]→[0,π] be defined by f(x)=cos−1(cosx)Then, the number of points xϵ[0,4π] satisfying the equation f(x)=10−x10, is ___
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Solution
(i) Using definition of f(x)=cos−1(x), we trace the curve f(x)=cos−1(cosx).
(ii) The number of solutions of equations involving trigonometric and algebraic functions and involving both functions are found using graphs of the curves
We know that,cos−1(cosx)=⎧⎪
⎪
⎪
⎪⎨⎪
⎪
⎪
⎪⎩x,Ifxϵ[0,π]2π−x,Ifxϵ[π,2π]−2π+x,Ifxϵ[2π,3π]4π−x,Ifxϵ[3π,4π]
For above graph, it is clear that y=10−x10 and y=cos−1(cosx) intersect at three distinct points, so number of solutions is 3.