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Question Number 48361 by peter frank last updated on 22/Nov/18

$$\mathrm{Solve}\mathrm{the}\mathrm{equation}$$$${\cos }^{-1}\mathrm{x}+{\cos }^{-1}(\frac{\mathrm{x}}{2}+\frac{\sqrt{3-{3\mathrm{x}}^2}}{2})=\frac{\pi }{3}$$

Commented by MJS last updated on 22/Nov/18

$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{true}\mathrm{for}x⩾\frac{1}{2}$$

Answered by behi83417@gmail.com last updated on 22/Nov/18

$$thisisnotanequation.$$$$let:a={cos}^{-1}x,b={cos}^{-1}(\frac{x}{2}+\frac{\sqrt{3-3x^2}}{2})$$$$cosb=\frac{x}{2}+\frac{\sqrt{3-3x^2}}{2}=\frac{1}{2}.x+\frac{\sqrt{3}}{2}.\sqrt{1-x^2}=$$$$=cos\frac{\pi }{3}cosa+sin\frac{\pi }{3}.sina=cos(\frac{\pi }{3}-a)$$$$\Rightarrow b=\frac{\pi }{3}-a\Rightarrow a+b=\frac{\pi }{3},for:x\in [-1,1].$$

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