Question

How do you solve \[2\tan x = \sin 2x\] ?

Answer 1

Hint:To solve \[2\tan x = \sin 2x\], we will first write \[\tan x\] in terms of \[\sin x\] and \[\cos x\]. As we know \[\tan x = \dfrac{{\sin x}}{{\cos x}}\] and from double angle trigonometric formula we can write \[\sin 2x = 2\sin x\cos x\], using this we will rewrite the given equation. Then we will simplify it and use the general solution condition to find the result.

Complete step by step answer:
Given equation is \[2\tan x = \sin 2x - - - (1)\].
As we know from the double angle formula: \[\sin 2x = 2\sin x\cos x\].
Also, we know that \[\tan x = \dfrac{{\sin x}}{{\cos x}}\].
On putting this value in \[(1)\], we get
\[ \Rightarrow 2\dfrac{{\sin x}}{{\cos x}} = 2\sin x\cos x\]
Taking all the terms to the left-hand side of the equation, we get
\[ \Rightarrow 2\dfrac{{\sin x}}{{\cos x}} - 2\sin x\cos x = 0\]
Taking \[2\] common, we get
\[ \Rightarrow 2\left( {\dfrac{{\sin x}}{{\cos x}} - \sin x\cos x} \right) = 0\]
Dividing both the sides by \[2\], we get
\[ \Rightarrow \dfrac{{\sin x}}{{\cos x}} - \sin x\cos x = 0\]

Taking \[\sin x\] common, we get
\[ \Rightarrow \sin x\left( {\dfrac{1}{{\cos x}} - \cos x} \right) = 0\]
On simplifying, we get
\[ \Rightarrow \dfrac{{\sin x\left( {1 - {{\cos }^2}x} \right)}}{{\cos x}} = 0\]
On solving, we get
\[ \Rightarrow \sin x = 0\] or \[1 - {\cos ^2}x = 0\]
\[ \Rightarrow \sin x = 0\] or \[{\cos ^2}x = 1\]
On further simplification, we get
\[ \Rightarrow \sin x = 0\] or \[\cos x = \pm 1\]
General solution of \[\sin x = 0\] is \[x = n\pi \], where \[n \in Z\]. Also, general solution of \[\cos x = 1\] is \[x = 2n\pi \], where \[n \in Z\]and general solution of \[\cos x = - 1\] is \[x = \left( {2n + 1} \right)\pi \], where \[n \in Z\]. On combining, the general solution of \[\cos x = \pm 1\] is \[x = n\pi \], where \[n \in Z\].

Therefore, the solution of \[2\tan x = \sin 2x\] is \[x = n\pi \] where \[n \in Z\].

Note:An equation involving two 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 whereas a trigonometric equation is satisfied for some particular values of the unknown angles.