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Find all solutions to the equation in the interval [0,2pi). Enter the solutions in increasing order. Cos 2x = cos x

Find All Solutions To The Equation In The Interval 02pi Enter The Solutions In Increasing Order Cos 2x Cos X class=

Sagot :

Step-by-step explanation:

[tex] \cos(2x) = \cos(x) [/tex]

[tex] \cos {}^{2} (x) - \sin {}^{2} (x) = \cos(x) [/tex]

[tex] \cos {}^{2} (x) - (1 - \cos {}^{2} (x) ) = \cos(x) [/tex]

[tex]2 \cos {}^{2} (x) - 1 = \cos(x) [/tex]

[tex]2 \cos {}^{2} (x) - \cos(x) - 1 = 0[/tex]

[tex]2 \cos {}^{2} (x) - 2 \cos(x) + \cos(x) - 1 = 0[/tex]

[tex]2 \cos(x) ( \cos(x) - 1) + 1( \cos(x) + 1)[/tex]

[tex](2 \cos(x) + 1)( \cos(x) - 1) = 0[/tex]

[tex]2 \cos(x) + 1 = 0[/tex]

[tex]2 \cos(x) = - 1[/tex]

[tex] \cos(x) = - \frac{1}{2} [/tex]

[tex]x = \frac{2\pi}{3} ,x = \frac{4\pi}{3} [/tex]

[tex] \cos(x) - 1 = 0[/tex]

[tex] \cos(x) = 1[/tex]

[tex]x = 0[/tex]

So our answer are

[tex]0, \frac{2\pi}{3} , \frac{4\pi}{3} [/tex]

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