Question

How do you solve for all solutions of x in $\sin 2x - 1 = 0?$

Answer 1

Hint: To solve for the given trigonometric equation, first consider the argument of the given trigonometric solution to be some variable (say $\theta $) and then solve the given equation for that trigonometric function first, after that find the value of the argument by taking inverse of the respective trigonometric function in both sides of the equation. Finally convert the assumed argument to the original one and find the required solution.

Complete step by step solution:
In order to solve the given trigonometric equation $\sin 2x - 1 = 0$, we will first consider the argument of sine function, that is $2x$ to be $\theta $, then the given trigonometric equation can be written as $\sin \theta - 1 = 0$
Now solving this equation for sine function, we will get
$
\Rightarrow \sin \theta - 1 = 0 \\
\Rightarrow \sin \theta = 1 \\
$
Taking both sides the inverse function of sine,
$ \Rightarrow {\sin ^{ - 1}}\left( {\sin \theta } \right) = {\sin ^{ - 1}}\left( 1 \right)$
We can further write it as follows
$ \Rightarrow \theta = {\sin ^{ - 1}}\left( 1 \right)$
Now from table of basic values of inverse trigonometric functions, we know that ${\sin ^{ - 1}}\left( 1
\right) = \dfrac{\pi }{2}$
$ \Rightarrow \theta = \dfrac{\pi }{2}$
But this is the principle value for the given trigonometric equation, and principle solutions lie in the range of $\left[ {0,\;2\pi } \right]$ which do not cover all other possible solutions.
So we will find its general solution, since sine function is a periodic function and repeat its values after an interval of $2\pi $, so from this we can write the general solution as
$ \Rightarrow \theta = 2n\pi \pm \dfrac{\pi }{2},\;{\text{where}}\;n \in I$
Now replacing $\theta \;{\text{with}}\;2x$
$
\Rightarrow 2x = 2n\pi \pm \dfrac{\pi }{2} \\
\Rightarrow x = n\pi \pm \dfrac{\pi }{4},\;{\text{where}}\;n \in I \\
$

Note: Apart from principle solution and general solution, there is one more solution that exists for periodic functions, which is known as a particular solution. And in particular solutions, we find solutions of the given periodic equation in a particular range.