
How do you solve for \[x\] in \[3\sin 2x = \cos 2x\] for the interval \[0 \leqslant x < 2\pi \]?
Answer
558.6k+ views
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,
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 |
| \[\sin x = 0\] | \[x = n\pi \] |
| \[\cos x = 0\] | \[x = n\pi + \dfrac{\pi }{2}\] |
| \[\tan x = 0\] | \[x = n\pi \] |
| \[\sin x = \sin \alpha \] | \[x = n\pi \pm {\left( { - 1} \right)^n}\alpha \] |
| \[\cos x = \cos \alpha \] | \[x = 2n\pi \pm \alpha \] |
| \[\tan x = \tan \alpha \] | \[x = n\pi \pm \alpha \] |
Recently Updated Pages
Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Maths: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Master Class 11 Social Science: Engaging Questions & Answers for Success

Master Class 11 Physics: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Trending doubts
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Discuss the various forms of bacteria class 11 biology CBSE

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE

State the laws of reflection of light

Explain zero factorial class 11 maths CBSE

10 examples of friction in our daily life

