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

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To solve the equation cos 6x = 1/2, the principal values of 6x are found at angles where cos equals 1/2, specifically at π/3 and 5π/3. This leads to multiple solutions for x within the range of 0 to 2π, including values such as π/18, 5π/18, and 17π/18. Converting these radian measures to degrees results in corresponding angles of 10°, 50°, and 170°. The complete set of solutions in degrees includes 10°, 50°, 70°, 110°, 130°, 170°, 190°, 230°, 250°, 290°, 310°, and 350°. The discussion emphasizes the importance of converting radian measures to degrees for clarity.
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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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