Find all values of x in the interval $[0,2\pi ]$ that satisfy the equation:

$2+\mathrm{cos}2x=3\mathrm{cos}x$

$2+\mathrm{cos}2x=3\mathrm{cos}x$

Rapsinincke
2022-07-16
Answered

Find all values of x in the interval $[0,2\pi ]$ that satisfy the equation:

$2+\mathrm{cos}2x=3\mathrm{cos}x$

$2+\mathrm{cos}2x=3\mathrm{cos}x$

You can still ask an expert for help

Ronald Hickman

Answered 2022-07-17
Author has **18** answers

$2+\mathrm{cos}2x=3\mathrm{cos}x$

$\mathrm{cos}2x=2{\mathrm{cos}}^{2}x-1$

$=1-2{\mathrm{sin}}^{2}x$

$2+2{\mathrm{cos}}^{2}x-1=3\mathrm{cos}x$

$2{\mathrm{cos}}^{2}x-1=3\mathrm{cos}x$

$2{\mathrm{cos}}^{2}x-3\mathrm{cos}x+1=0$

$(2\mathrm{cos}x-1)(\mathrm{cos}x-1)=0$

$\mathrm{cos}x=\frac{1}{2}\text{}\text{}\text{}\mathrm{cos}x=1$

$x=\frac{\pi}{3};\frac{5\pi}{3}\text{}\text{}\text{}x=0;2\pi $

$\mathrm{cos}2x=2{\mathrm{cos}}^{2}x-1$

$=1-2{\mathrm{sin}}^{2}x$

$2+2{\mathrm{cos}}^{2}x-1=3\mathrm{cos}x$

$2{\mathrm{cos}}^{2}x-1=3\mathrm{cos}x$

$2{\mathrm{cos}}^{2}x-3\mathrm{cos}x+1=0$

$(2\mathrm{cos}x-1)(\mathrm{cos}x-1)=0$

$\mathrm{cos}x=\frac{1}{2}\text{}\text{}\text{}\mathrm{cos}x=1$

$x=\frac{\pi}{3};\frac{5\pi}{3}\text{}\text{}\text{}x=0;2\pi $

Jeffrey Jordon

Answered 2022-08-01
Author has **2313** answers

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