SATHEE: Inverse Circular Functions 2 Question 15

Inverse Circular Functions 2 Question 15

15. If $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = \cos^{- 1} (\cos x)$ . Then, the number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ , is

(2014 Adv.)

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Solution:

PLAN

(i) Using definition of $f (x) = \cos^{- 1} (x)$ , we trace the curve $f (x) = \cos^{- 1} (\cos x)$

(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, $\cos^{- 1} (\cos x) = \begin{array}{ll} x, & if x \in [0, π] 2 π - x, & if x \in [π, 2 π] - 2 π + x, & if x \in [2 π, 3 π] 4 π - x, & if x \in [3 π, 4 π] \end{array}$

From above graph, it is clear that $y = \frac{10 - x}{10}$ and $y = \cos^{- 1} (\cos x)$ intersect at three distinct points, so number of solutions is 3 .

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Inverse Circular Functions 2 Question 15

15. If f:[0,4π]→[0,π] be defined by f(x)=cos−1⁡(cos⁡x). Then, the number of points x∈[0,4π] satisfying the equation f(x)=10−x10, is

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15. If $f : [0, 4 π] \to [0, π]$ be defined by $f (x) = \cos^{- 1} (\cos x)$ . Then, the number of points $x \in [0, 4 π]$ satisfying the equation $f (x) = \frac{10 - x}{10}$ , is