Question

Find the principle solutions of $\cos x = \dfrac{1}{2}$

Answer 1

In order to solve this question we will first learn the meaning of principle solutions then we will find the value of x in in the first quadrant where it is equal to $\dfrac{1}{2}$ and in the other quadrant of the same value so like this we will find the two solutions of this questions which will be principle solutions.

Complete step by step solution:
For solving this question we will first learn what are the principle solutions:
So the principle solution is the value of solutions in between 0 to $2\pi$ .
So we will find the first value of $\cos x = \dfrac{1}{2}$ will be at $\dfrac{\pi }{3}$
Now for the second on as we know that the value of $\cos x$ is positive at 1st and 4th quadrant so we will put it as
$\cos \left( {2\pi - \dfrac{\pi }{3}} \right) = \dfrac{1}{2}$
So on further solving it we will get the value of
$\cos \left( {\dfrac{{5\pi }}{3}} \right) = \dfrac{1}{2}$
Such that $0 < \dfrac{\pi }{3} < 2\pi$ and $0 < \dfrac{{5\pi }}{3} < 2\pi$
So these are the two principle solutions of these questions.
So, the correct answer is “ $0 < \dfrac{\pi }{3} < 2\pi$ and $0 < \dfrac{{5\pi }}{3} < 2\pi$ ”.

Note: While solving these types of questions we should always keep in mind that there will be always two angles for the principle solutions of these questions. One angle will be directly found through the first quadrant and its complementary angle. So like this we will solve these questions