Question

How do you solve $\sin \left( {2x} \right) = \dfrac{1}{2}$ ?

Answer 1

The question belongs to the solution of trigonometric equations. In this question we will convert the right hand side of the equation in trigonometric ratio. For this we will take $\sin$ to the right hand side of the equation by keeping in mind that the original trigonometric equation should not be changed. We will find the angle from the standard trigonometric angle ratio table so that $\sin$ of the respective angle gives the appropriate value as required. Then we will solve the equation for the value of the trigonometric ratio.

Complete step by step solution:
Step: 1 the given trigonometric equation is,
$\sin (2x) = \dfrac{1}{2}$
Take $\sin$ to the right hand side of the equation.
$\Rightarrow \sin 2x = \sin 30$
We know that the value of $\sin 30 = \dfrac{1}{2}$ .
Step: 2 now compare the both side of the equation to find the value of $x$ .
$\Rightarrow 2x = 30$
Solve the linear equation to find the value of $x$.
Divide both side of the equation by two to find the value of $x$ .
$\Rightarrow 2x = 30 \\\ \Rightarrow \dfrac{{2x}}{2} = \dfrac{{30}}{2} \\\ \Rightarrow x = 15 \\\$
Step: 3 now consider the other angle at which $\sin$ gives the value as required.
We know that the value of $\sin 150 = \dfrac{1}{2}$.
Now again take $\sin$ of both side of the equation.
$\Rightarrow \sin 2x = \sin 150$
Compare the both side of the equation to solve the given trigonometric equation.
$\Rightarrow 2x = 150$
Solve the linear equation to find the value of $x$ .
$\Rightarrow 2x = 150 \\\ \Rightarrow x = \dfrac{{150}}{2} \\\ \Rightarrow x = 75 \\\$
Final Answer:
Therefore the solution of the given equation is $x = 15$ and $x = 75$ .

Note:
You are advised to remember the table of standard trigonometric ratios of angle. They must know to solve the linear equation. They should solve the given equation by taking $\sin$ to both sides of the equation.