find all solutions of the equation.\n\n\\( \\cos x=-\\frac{1}{2} \\)\n\nselect the correct choice below and…

find all solutions of the equation.\n\n\\( \\cos x=-\\frac{1}{2} \\)\n\nselect the correct choice below and, if necessary, fill in the ar\n\n\\( \\bigcirc \\mathrm{a} \\). \\( x= \\)\n(simplify your answer. type your answer(s) as \n expression. use a comma to separate answe\n\n\\( \\bigcirc \\mathrm{b} \\). there is no solution.

find all solutions of the equation.\n\n\\( \\cos x=-\\frac{1}{2} \\)\n\nselect the correct choice below and, if necessary, fill in the ar\n\n\\( \\bigcirc \\mathrm{a} \\). \\( x= \\)\n(simplify your answer. type your answer(s) as \n expression. use a comma to separate answe\n\n\\( \\bigcirc \\mathrm{b} \\). there is no solution.

Answer

Explanation:

Step1: Recall the unit - circle values

We know that (\cos x =-\frac{1}{2}). On the unit circle, (\cos\theta=x) coordinate of the point on the unit circle corresponding to the angle (\theta). The general solution of the equation (\cos x = a) is (x = 2k\pi\pm\cos^{- 1}(a)), where (k\in\mathbb{Z}). Since (\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}), for (\cos x=-\frac{1}{2})

Step2: Find the principal solutions

The principal solutions in the interval ([0,2\pi]) are (x = \frac{2\pi}{3}) (because (\cos\left(\frac{2\pi}{3}\right)=-\frac{1}{2})) and (x=\frac{4\pi}{3}) (because (\cos\left(\frac{4\pi}{3}\right)=-\frac{1}{2}))

Step3: Write the general solution

The general solution of the equation (\cos x=-\frac{1}{2}) is (x = 2k\pi\pm\frac{2\pi}{3}), where (k\in\mathbb{Z})

Answer:

(x = 2k\pi\pm\frac{2\pi}{3},k\in\mathbb{Z})