Question

Solve the following trigonometric expression to find the general value of $x$ :
$\cos x=\dfrac{1}{2}$.

Answer 1

Hint: Find the value of the cosine of the angle for which its value is $\dfrac{1}{2}$. Assume the angle as $\theta$ and write the above equation in the form of, $\cos x=\cos \theta$. Use the relation for the general solution of the cosine function, given by: $x=2n\pi \pm \theta$, where n belongs to any integer.

Complete step-by-step answer:
Trigonometric equations are equations involving trigonometric functions. A trigonometric equation that holds true for any angle is called a trigonometric identity. There are other equations that are true for certain angles. They are generally known as conditional equations.
Now, when we talk about the solutions of trigonometric equations, there are two types of solutions: general solution and principal solution. There are an infinite number of positive and negative angles that satisfy an equation. We cannot write all of them, so we generalize them using an integer so that one can find any solution by just putting the value of integer. However, in principle there is a limit given to us between which we are required to find the solution. Here we are going to use the sum to product rule.
Now, let us come to the question. We have been given: $\cos x=\dfrac{1}{2}$.
We know that, $\cos \dfrac{\pi }{3}=\dfrac{1}{2}$, therefore the above equation becomes,
$\cos x=\cos \dfrac{\pi }{3}$
Using the relation for general solution of cosine function, given by: $x=2n\pi \pm \theta$, we get,
$x=2n\pi \pm \dfrac{\pi }{3}$
Taking $\dfrac{\pi }{3}$ common, we get,
$x=\dfrac{\pi }{3}\left( 6n\pm 1 \right)$, ‘n’ belongs to integers.

Note: One may note that we have found the general solution of the given equation because we haven’t been provided with any particular limit of ‘x’ in which we have to find the solution, so that means we have to generalize the solution. By putting the integral values of ‘n’ we can find an infinite number of solutions of the given equation.