Question

Solve $\cos \theta = \dfrac{1}{2}$

Answer 1

We know that the two dimensional planes are divided into 4 quadrants as given below:

Quadrant I:$0\; - \;\dfrac{\pi }{2}$ All values are positive.
Quadrant II:$\dfrac{\pi }{2}\; - \;\pi$ Only Sine and Cosec values are positive.
Quadrant III:$\pi \; - \;\dfrac{{3\pi }}{2}$ Only Tan and Cot values are positive.
Quadrant IV:$\dfrac{{3\pi }}{2}\; - \;2\pi$ Only Cos and Sec values are positive.
Now using this basic knowledge we can find the quadrant where cosine would be positive.
Also in order to solve $\cos \theta = \dfrac{1}{2}$we have to find the value of$\theta$.

Complete step by step solution:
Given
$\cos \theta = \dfrac{1}{2}...........................\left( i \right)$
Now we have to solve the equation (i), for that we have to find the value of$\theta$. We know that the trigonometric values repeat after a certain interval and since cosine is a trigonometric function we have to find the intervals in which it repeats and thereby the different values of $\theta$.
On observing (i) we know can say from the standard formula that:
$\theta = \dfrac{\pi }{3}........................\left( {ii} \right)$
Now we know that the cosine value is positive in Quadrant I and Quadrant IV so to find the next value we have to subtract the value given in (ii) with $2\pi$to find the angle in the IV Quadrant.
i.e.

$$\theta = 2\pi - \dfrac{\pi }{3} \\\ \theta = \dfrac{5}{3}\pi .......................\left( {iii} \right) \\\$$

Also now we know that the period in which cosine repeats is $2\pi$which will repeat in both directions of$2\pi$.

Such that in general we can write:
$\theta = \dfrac{\pi }{3} + 2\pi n\,\,{\text{and}}\,\,\dfrac{{5\pi }}{3} + 2\pi n \\\ n:{\text{any}}\,{\text{integer}} \\\$
Note: cosine values of some general angles are given below:

$$\theta \;\;\;\;\;\;\;\cos \theta \\\ \dfrac{\pi }{3}\;\;\;\;\;\;\;\;\dfrac{1}{2} \\\ \dfrac{\pi }{6}\;\;\;\;\;\;\;\dfrac{{\sqrt 3 }}{2} \\\ \dfrac{\pi }{4}\;\;\;\;\;\;\;\dfrac{1}{{\sqrt 2 }} \\\$$

While approaching a trigonometric problem one should keep in mind that one should work with one side at a time and manipulate it to the other side.