Question

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

Answer 1

Hint: 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
\[\sin x = 0\]	\[x = n\pi \]
\[\cos x = 0\]	\[x = n\pi + \dfrac{\pi }{2}\]
\[\tan x = 0\]	\[x = n\pi \]
\[\sin x = \sin \alpha \]	\[x = n\pi \pm {\left( { - 1} \right)^n}\alpha \]
\[\cos x = \cos \alpha \]	\[x = 2n\pi \pm \alpha \]
\[\tan x = \tan \alpha \]	\[x = n\pi \pm \alpha \]