Solve cos 6x=(1/2) for principal values in degree

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SUMMARY

The equation cos 6x = (1/2) yields principal values for x in the interval 0 ≤ x < 2π. The solutions for 6x are derived from the angles where cos θ = (1/2), specifically at θ = π/3 and θ = 5π/3, leading to multiple values for x: π/18, 5π/18, 7π/18, 11π/18, 13π/18, 17π/18, 19π/18, 23π/18, 25π/18, 29π/18, 31π/18, and 35π/18. Converting these radian measures to degrees results in corresponding values of 10°, 50°, 70°, 110°, 130°, 170°, 190°, 230°, 250°, 290°, 310°, and 350°.

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cos 6x=(1/2)
 
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blake said:
cos 6x=(1/2)

note $\cos{\theta} = \dfrac{1}{2}$ at $\theta = \dfrac{\pi}{3} \text{ and } \dfrac{5\pi}{3}$

$0 \le x < 2\pi \implies 0 \le 6x < 12\pi$

$\cos(6x) = \dfrac{1}{2} \implies 6x = \dfrac{\pi}{3} \, , \, \dfrac{5\pi}{3} \, , \, \dfrac{7\pi}{3} \, , \, \dfrac{11\pi}{3} \, , \, \dfrac{13\pi}{3} \, , \, \dfrac{17\pi}{3}\, , \, \dfrac{19\pi}{3} \, , \, \dfrac{23\pi}{3} \, , \, \dfrac{25\pi}{3} \, , \, \dfrac{29\pi}{3} \, , \, \dfrac{31\pi}{3} \, , \, \dfrac{35\pi}{3}$

$x = \dfrac{\pi}{18} \, , \, \dfrac{5\pi}{18} \, , \, \dfrac{7\pi}{18} \, , \, \dfrac{11\pi}{18} \, , \, \dfrac{13\pi}{18} \, , \, \dfrac{17\pi}{18}\, , \, \dfrac{19\pi}{18} \, , \, \dfrac{23\pi}{18} \, , \, \dfrac{25\pi}{18} \, , \, \dfrac{29\pi}{18} \, , \, \dfrac{31\pi}{18} \, , \, \dfrac{35\pi}{18}$
 
Skeeter's answer is, of course, in radians. To get the answer in degrees remember that \pi radians is 180 degrees. That is, \frac{180}{\pi}= 1 so \frac{\pi}{18} radians is the same as \frac{\pi}{18}\frac{180}{\pi}= 10 degrees.
 

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