Question

Why sometimes in the trigonometric equations in the answers equation in the answers we add $2n \pi$ and other times only $n \pi$?

Answer 1

It all depends on which trigonometric function we are dealing with, because different trigonometric functions have different periods. For sine and cosine functions we have a period of $2\pi$ and for tangent function we have a period of $\pi$.

Complete answer:
It depends on the trigonometric functions that we’re dealing with.
The period of $\sin x$ and $\cos x$ is $2\pi$, so if we are to find the general solutions to some equation like $\sin x = 1$, we’d have the solution in the form of $x = \dfrac{\pi }{2} \pm 2\pi n$.
Therefore, where we are dealing with functions like $\cos x$ and $\sin x$we had to add $2nPi$.
The period of $\tan x$, on the other hand, is $\pi$, so an equation like $\tan x = 1$ would have the general solution $x = \dfrac{\pi }{4} \pm \pi n$.
Therefore, where we are dealing with functions like $\tan x$ we had to add $nPi$.

Note: We know that sin x and cos x repeat themselves after an interval of 2π, and tan x repeats itself after an interval of π. The solutions such as trigonometry equations which lie in the interval of [0, 2π] are called principal solutions. A trigonometric equation will also have a general solution expressing all the values which would satisfy the given equation, and it is expressed in a generalized form in terms of ‘n’. Since sine, cosine and tangent are the major trigonometric functions, hence the solutions will be derived for the equations comprising these three ratios. However, the solutions for the other three ratios such as secant, cosecant and cotangent can be obtained with the help of those solutions.