Question

How do you solve for $x$ in $3\sin 2x = \cos 2x$ for the interval $0 \leqslant x < 2\pi$?

Answer 1

In this question we have to solve the trigonometric equation to get the values for $x$, first we will transform the equation in terms of $\tan x$ by using the trigonometric identity $\dfrac{{\sin x}}{{\cos x}} = \tan x$, and now using the general solution for the $\tan x$ function which is given by, $n\pi + x$, where $n \in Z$, now substituting different values for $n$ to get the required values in the interval $0 \leqslant x < 2\pi$.

Complete step-by-step answer:
Given equation is $3\sin 2x = \cos 2x$.
Now divide both sides with 3, we get,
$\Rightarrow \dfrac{{3\sin 2x}}{3} = \dfrac{{\cos 2x}}{3}$,
Now simplifying we get,
$\Rightarrow \sin 2x = \dfrac{{\cos 2x}}{3}$,
Now again divide both sides with $\cos 2x$, we get,
$\Rightarrow \dfrac{{\sin 2x}}{{\cos 2x}} = \dfrac{{\cos 2x}}{{3\cos 2x}}$,
Now simplifying we get,
$\Rightarrow \dfrac{{\sin 2x}}{{\cos 2x}} = \dfrac{1}{3}$,
Now using the trigonometric identity,$\dfrac{{\sin x}}{{\cos x}} = \tan x$, we get,
$\Rightarrow \tan 2x = \dfrac{1}{3}$,
Now we know that the general solution for $\tan x$ will be given as,$n\pi + x$, where $n \in Z$, now using this fact we get,
$\Rightarrow 2x = n\pi + {\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right)$,
Now substituting the value of ${\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right) = 0.3217$ in the above equation we get,
$\Rightarrow 2x = n\pi + 0.3217$,
Now dividing both sides with 2 we get,
$\Rightarrow \dfrac{{2x}}{2} = \dfrac{{n\pi + 0.3217}}{2}$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{n\pi }}{2} + \dfrac{{0.3217}}{2}$,
Now again simplifying we get,
$\Rightarrow x = \dfrac{{n\pi }}{2} + 0.1608$,
So, now the given interval is equal to $0 \leqslant x < 2\pi$, now taking different values for $n$, and substituting the values in the above equation we get,
First take $n = 0$, as the given interval is from 0,
$\Rightarrow x = \dfrac{{\left( 0 \right)\pi }}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = 0.1608$, and it lies between the given interval,
First take $n = 1$, as the given interval is from 0,
$\Rightarrow x = \dfrac{{\left( 1 \right)\pi }}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{3.14}}{2} + 0.1608$
Again simplifying we get,
$\Rightarrow x = 1.57 + 0.1608$
Now simplifying by adding we get,
$\Rightarrow x = 1.7308$, and it lies between the given interval,
Now take $n = 2$, as the given interval is from 0,
$\Rightarrow x = \dfrac{{\left( 2 \right)\pi }}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = 3.14 + 0.1608$
Now simplifying by adding we get,
$\Rightarrow x = 3.302$, and it lies between the given interval,
Now take $n = 3$, as the given interval is from 0,
$\Rightarrow x = \dfrac{{\left( 3 \right)\pi }}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{3\left( {3.14} \right)}}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{9.42}}{2} + 0.1608$,
Now dividing and simplifying we get,
$\Rightarrow x = 4.71 + 0.1608$,
Now simplifying by adding we get,
$\Rightarrow x = 4.8708$, and it lies between the given interval,
Now take $n = 4$, as the given interval is from 0,
$\Rightarrow x = \dfrac{{\left( 4 \right)\pi }}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{4\left( {3.14} \right)}}{2} + 0.1608$,
Now simplifying we get,
$\Rightarrow x = \dfrac{{12.56}}{2} + 0.1608$,
Now dividing and simplifying we get,
$\Rightarrow x = 6.28 + 0.1608$,
Now simplifying by adding we get,
$\Rightarrow x = 6.4408$, and it doesn’t lies between the given interval,
So the possible values of $x$ for the given equation are, 0.1608, 1.7308, 3.302, and 4.8708.

**The possible values of $x$ in $3\sin 2x = \cos 2x$ for the interval $0 \leqslant x < 2\pi$ are 0.1608, 1.7308, 3.302, and 4.8708. **

Note:
An equation involving one or more trigonometric ratios of an unknown angle is called a trigonometric equation. A trigonometric equation is different from a trigonometric identity. An identity is satisfied for every value of the unknown angle, and a trigonometric equation is satisfied for some particular values of the unknown angle. A value of the unknown angle which satisfies the trigonometric equation is called its solution. Here are some general solutions for some trigonometric equations,

Trigonometric equation	General solutions
$\sin x = 0$	$x = n\pi$
$\cos x = 0$	$x = n\pi + \dfrac{\pi }{2}$
$\tan x = 0$	$x = n\pi$
$\sin x = \sin \alpha$	$x = n\pi \pm {\left( { - 1} \right)^n}\alpha$
$\cos x = \cos \alpha$	$x = 2n\pi \pm \alpha$
$\tan x = \tan \alpha$	$x = n\pi \pm \alpha$