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Answer 1 · 2017-08-14T22:23:08

$x = \frac{π}{3}, \frac{5 π}{3}$ $x = frac(pi)(3), frac(5 pi)(3)$

We have: $cos (x) = \frac{1}{2}$ $cos(x) = frac(1)(2)$ ; $0 \leq x \leq 2 π$ $0 le x le 2 pi$

Let the reference angle be $cos (x) = \frac{1}{2}$ $cos(x) = frac(1)(2)$ :

Applying $arccos$ $arccos$ to both sides of the equation:

$\Rightarrow arccos (cos (x)) = arccos (\frac{1}{2})$ $Rightarrow arccos(cos(x)) = arccos(frac(1)(2))$

$\Rightarrow x = \frac{π}{3}$ $Rightarrow x = frac(pi)(3)$

So, the reference angle is $x = \frac{π}{3}$ $x = frac(pi)(3)$

Now, the interval is given as $0 \leq x \leq 2 π$ $0 le x le 2 pi$ , covering all four quadrants.

The value of $cos (x)$ $cos(x)$ is positive, i.e. $+ \frac{1}{2}$ $+ frac(1)(2)$ .

So, we need to find the values of $x$ $x$ in the first and fourth quadrants (where values of $cos (x)$ $cos(x)$ are positive).

$\Rightarrow x = \frac{π}{3}, 2 π - \frac{π}{3}$ $Rightarrow x = frac(pi)(3), 2 pi - frac(pi)(3)$

$\Rightarrow x = \frac{π}{3}, \frac{6 π}{3} - \frac{π}{3}$ $Rightarrow x = frac(pi)(3), frac(6 pi)(3) - frac(pi)(3)$

$therefore x = frac(pi)(3), frac(5 pi)(3)$

Therefore, the solutions to the equation are $x = frac(pi)(3)$ and $x = frac(5 pi)(3)$ .