How to do this problem?

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1 Answer
Aug 14, 2017

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

Explanation:

We have: cos(x) = frac(1)(2); 0 le x le 2 pi

Let the reference angle be cos(x) = frac(1)(2):

Applying arccos to both sides of the equation:

Rightarrow arccos(cos(x)) = arccos(frac(1)(2))

Rightarrow x = frac(pi)(3)

So, the reference angle is x = frac(pi)(3)

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

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

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

Rightarrow x = frac(pi)(3), 2 pi - frac(pi)(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).